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2012

Neutrosophic Super Matrices and Quasi Super Matrices

Florentin Smarandache University of New Mexico, [email protected]

W.B. Vasantha Kandasamy [email protected]

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Recommended Citation W.B. Vasantha Kandasamy & F. Smarandache. Neutrosophic Super Matrices and Quasi Super Matrices. Ohio: Zip Publishing, 2012.

This Book is brought to you for free and open access by the Mathematics at UNM Digital Repository. It has been accepted for inclusion in Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected].

Neutrosophic Super Matrices and Quasi Super Matrices

W. B. Vasantha Kandasamy Florentin Smarandache

ZIP PUBLISHING Ohio 2012 This book can be ordered from:

Zip Publishing 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Toll Free: (614) 485-0721 E-mail: [email protected] Website: www.zippublishing.com

Copyright 2012 by Zip Publishing and the Authors

Peer reviewers:

Prof. Valeri Kroumov, Okayama Univ. of Science, Japan. Dr.S.Osman, Menofia University, Shebin Elkom, Egypt. Dr. Sebastian Nicolaescu, 2 Terrace Ave., West Orange, NJ 07052, USA. Prof. Gabriel Tica, Bailesti College, Bailesti, Jud. Dolj, Romania.

Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm

ISBN-13: 978-1-59973-194-0 EAN: 9781599731940

Printed in the United States of America

2

CONTENTS

Preface 5

Chapter One INTRODUCTION 7

Chapter Two NEUTROSOPHIC SUPER MATRICES 9

Chapter Three SUPER BIMATRICES AND THEIR GENERALIZATION 87

3.1 Super Bimatrices and their Properties 87 3.2 Operations on Super Bivectors 139

3

Chapter Four QUASI SUPER MATRICES 157

FURTHER READING 195

INDEX 197

ABOUT THE AUTHORS 200

4

PREFACE

In this book authors study neutrosophic super matrices. The concept of neutrosophy or indeterminacy happens to be one the powerful tools used in applications like FCMs and NCMs where the expert seeks for a neutral solution. Thus this concept has lots of applications in fuzzy neutrosophic models like NRE, NAM etc. These concepts will also find applications in image processing where the expert seeks for a neutral solution. Here we introduce neutrosophic super matrices and show that the sum or product of two neutrosophic matrices is not in general a neutrosophic super matrix. Another interesting feature of this book is that we introduce a new class of matrices called quasi super matrices; these matrices are the larger class which contains the class of super matrices. These class of matrices lead to more partition of n × m matrices where n > 1 and m > 1, where m and n can also be equal. Thus this concept cannot be defined on usual row matrices or column matrices. These matrices will play a major role when studying a problem which needs multi fuzzy neutrosophic models.

5 This book is organized into four chapters. Chapter one is introductory in nature. Chapter two introduces the notion of neutrosophic super matrices and describes operations on them. In chapter three the major product of super row vectors and column vectors are defined and are extended to bisuper vectors. The final chapter introduces the new concept of quasi super matrices and suggests some open problems. We thank Dr. K.Kandasamy for proof reading and being extremely supportive.

W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE

6

Chapter One

INTRODUCTION

In this chapter we just indicate the concepts which we have used and the place of reference of the same.

Throughout this book I will be denote the indeterminate 2 such that I = I. Z  I, Q  I, R  I, C  I, Zn  I and C(Zn)  I denotes the neutrosophic ring [3-4, 7]. For the definition and properties of super matrices please refer [2, 5]. The notion of bimatrices is introduced in [6].

Here in this book the notion of neutrosophic super matrices are introduced and operations on them are performed.

We also can use the entries of these super neutrosophic matrice from rings; (Z  I) (g) = {a + bg | a, b  Z  I and g is the new element such that g2 = 0}. Z can be replaced by R or Q or Zn or C and we call them as dual number rings [11].

8 Neutrosophic Super Matrices and Quasi Super Matrices

(Q  I) (g1) = {a + bg1 | a, b  Q  I where g1 is a new 2 element such that g1 = g1}. We call (Q  I) (g1) as special dual like number neutrosophic rings. Here also Q can be replaced by R or Z or C or Zn.

(R  I) (g2) = {a + bg | a, b  R  I, g2 the new element 2 such that g2 = –g2} is the special quasi dual number neutrosophic ring. We can replace R by Q or Z or Zn or C and still we get special quasi dual number neutrosophic rings.

Finally we can also use (Z  I) (g1, g2, g3) = {a1 + a2g1 + a3g2 + a4g3 | ai  Z  I; 1  i  4; g1, g2, g3 are new elements 2 2 2 such that g1 = 0, g2 = g2 and g=3 –g3 with gigj = (0 or gi or gj), i  j; 1  i, j  3} the special mixed neutrosophic dual number rings [11-13].

Finally in this book the new notion of quasi super matrices is introduced and their properties are studied. This new concept of quasi super matrices can be used in the construction of fuzzy- neutrosophic models.

Chapter Two

NEUTROSOPHIC SUPER MATRICES

In this chapter we for the first time introduce and study the notion of super neutrosophic matrices and their properties.

A super matrix A in which the element indeterminacy I is present is called a super neutrosophic matrix. To be more precise we will define these notions.

DEFINITION 2.1: Let X = {(x1 x2 … xr | xr+1 … xt | xt+1 …. xn) where xi   R  I ; R reals 1 i  n) be a super row matrix which will be known as a super row neutrosophic matrix.

Thus all super row matrices in general are not super row neutrosophic matrix.

We illustrate this situation by an example.

Example 2.1: Let X = (3 I 1 –4 | 20 –5 | 2I 0 1 | 7 2 1 4), X is a super row neutrosophic matrix.

All super row matrices are not super row neutrosophic matrices.

10 Neutrosophic Super Matrices and Quasi Super Matrices

For take A = (3 1 0 1 1 | 25 – 27 | 0 1 2 3 4), A is only a super row matrix and is not a super row neutrosophic matrix.

Example 2.2: Let A = (I 3I 1 | –I 5I | 2I 0 3I 2 1 | 0 3); A is a super row neutrosophic matrix.

Example 2.3: Let A = (0.3I, 0, 1 | 0.7, 2 I, 0 I + 7 | –8I, 0.9I + 2); A is a super row neutrosophic matrix.

DEFINITION 2.2: Let

y1  y2   y r1 Y = where yi  R  I; 1  i  n.  

yt1    yn

Y is a super column matrix known as the super column neutrosophic matrix.

We illustrate this situation by some examples.

Example 2.4: Let

 2     I  34I  Y =    0   1     0.3I  Neutrosophic Super Matrices 11

be a super column matrix. Y is clearly a neutrosophic super column matrix (super neutrosophic column matrix).

Example 2.5: Let

 3I     2   0     1     Y =  5  ,  I     0.4     5   6I 

be a super neutrosophic column vector / matrix.

It is pertinent to mention here that in general all super column matrices are not super neutrosophic column matrices. This is proved by an example.

Example 2.6: Let

 3     0   1    31 A =   ,  4   0.5     2     1 

12 Neutrosophic Super Matrices and Quasi Super Matrices

be a super column matrix which is clearly not a super neutrosophic column matrix.

We now proceed onto define the notion of neutrosophic super square matrix.

DEFINITION 2.3: Let

aa11 12  a1n     aa21 22 a2n  A = (a ) =   ij        aan1 nn 

be a n  n super square matrix where aij  R  I. A is called the neutrosophic super square matrix of order n  n.

We illustrate this by an example.

Example 2.7: Let

 2I05 72    I130 I2I  01I3 I 7 A =   250 2 0 6  1353 I 7    111 I1 I

be a 66 super square matrix. Clearly as the entries of A are from R  I. A is neutrosophic super square matrix.

Neutrosophic Super Matrices 13

Example 2.8: Let

 30 I01 I5I    I1 015I01  23II00.323    A =  12 1I0I2I ,  34 2I23I34   5I 6 0 1 3 5I 6I    13I567 89I

A is a 77 neutrosophic square super matrix.

All super square matrices need not in general be a neutrosophic square super matrices.

We illustrate this by an example.

Example 2.9: Let

12345   67890 A = 09876   54321 24807

A is just a square super matrix and is not a square neutrosophic super matrix.

Now we proceed onto define a rectangular super matrix.

14 Neutrosophic Super Matrices and Quasi Super Matrices

DEFINITION 2.4: Let

 mmm11 12 13  m1n      mmm21 22 23 m2n  M = (mij) =       mmmp11 p12 p13 mp1n      mmmp1 p2 p3 mpn 

be a p  n (p  n) rectangular super matrix where mij  R  I. Clearly M is a rectangular neutrosophic p  n super matrix.

We illustrate this by an example.

Example 2.10: Let

30I654I3 2I1  520I23I4 5I6 A =  1I50I 23I45I  I0110 I 1 I 2

A is a 49 rectangular neutrosophic super matrix.

Example 2.11: Let

 3I120    I4I53I   123II4 A =   4I 5 6 0 2I  78I93 7     012I89I

be a 65 rectangular super neutrosophic matrix. Neutrosophic Super Matrices 15

All super rectangular matrices in general need not be a neutrosophic super rectangular matrices.

We illustrate this by an example.

Example 2.12: Let

12345678   90987654 A =   32101234   56789012

be a 4  8 super rectangular matrix. Clearly A is not a super neutrosophic rectangular matrix.

Thus we can say as in case of supermatrices if we take any neutrosophic matrix and partition it, we would get the super neutrosophic matrix.

We can as in case of supermatrices define a neutrosophic super matrix is one in which one or more of its elements are themselves simple neutrosophic matrices. The order of the neutrosophic super matrix is defined in the same way as that of a neutrosophic matrix.

We have to define addition and subtraction of neutrosophic super matrices.

We cannot add in general two neutrosophic matrices A and B of same natural order say m  n.

We shall first illustrate this by an example.

16 Neutrosophic Super Matrices and Quasi Super Matrices

Example 2.13: Let

356781 53I206  012I34   010I12 I0100I I0120I   A = 1234I0 and B = 0I60I0 0I1234 I23405   01I301 0123I0   0110I0 6I720I

be two super neutrosophic matrices of natural order 76. Clearly we cannot add A with B though as simple matrices A and B can be added.

Thus for two super matrices to have a compatible addition it is not sufficient they should be of same natural order they should necessarily be of same natural order but satisfy further conditions. This condition we shall define as matrix order of a super matrix.

DEFINITION 2.5: Let

 aa   11 1n  A =      aam1 mn 

be a neutrosophic super matrix where each aij is a simple neutrosophic matrix. 1  i  m and 1  j  n.

The natural order of A is t  s; clearly t  m and s  n that is A has t rows and s columns.

The matrix order of the neutrosophic super matrix A is m  n. i.e., it has m number of matrix rows and n number of Neutrosophic Super Matrices 17

matrix columns. The number rows in each simple neutrosophic matrix aip, i fixed is the same, p may vary; 1  p  n. Likewise for akj; j fixed, 1  k  m; the number columns is the same.

We now give examples of these.

Example 2.14: Let

3I12I102 I   02000I21 0 10I11I31 2 A =   2113I213 4 4I200I156I   II01010I3I

aaa  11 12 13  = aaa21 22 23    aaa31 32 33 

3I1 2I10 2I  where a11 = , a12 =   , a13 =   , 020 00I2 10

10I 11I3 12     a21 = 211, a22 = 3I21 , a23 = 34 , 4I2 00I1 56I

a31 = [I I 0], a32 = [1 0 1 0] and a33 = [I 3 I] are the neutrosophic submatrices of A also known as are the component matrices of A.

The natural order of A is 6  9 and the matrix order of A is 3  3. 18 Neutrosophic Super Matrices and Quasi Super Matrices

Example 2.15: Let

3I012  00I1 0 aa  11 12   A = 1112 3 = aa21 22   I0456I   aa31 32   1I0I 3

be super neutrosophic matrix. The natural order of A is 55 and the matrix order of A is 32.

Now having seen example of matrix order of a neutrosophic super matrix we now proceed on to define addition of two super neutrosophic matrices.

DEFINITION 2.6: Let A and B be two neutrosophic super matrices of same natural order say m  n. Let A and B be a neutrosophic super matrices of matrix order t  s where t  m and s  n.

We can say A+B the sum of the super neutrosophic matrices A and B be defined if and only if each simple matrix aij in A is of same order for each bij in B. 1  i  t; 1  j  s;

aa a bb b 11 12 1s  11 12 1s  i.e., if A =  and B =       aat1 t2 ats bbt1 t2 bts 

where aij and bij are simple matrices. Here order of aij = order of bij; 1 i  s and 1  j  t.

We illustrate this by some examples.

Neutrosophic Super Matrices 19

Example 2.16: Let

aaaa11 12 13 14 3I1 I A =  where a11 =   a12 =   , aaaa21 22 23 24 101 2

10 111I a13 = , a14 =   , a21 = [ 0 1 I ] , a22 = [3], 01 00I0 a23 = [1 I] and a24 =[5 1 I 0 ].

The natural order of A is 310.

3I0I10111I   A = 10120100I0 .   01I31I51I0

The matrix order of A is 24.

Take a neutrosophic super matrix B of natural order 310 and matrix order 24 given by

01I010101I  B = 10112101I0  I0121I1052

b11bbb 12 13 14 = . b21bbb 22 23 24

Since natural order of matrix aij = natural order of bij, for each i and j, 1  i  2 and 1  j  4 and as A and B have same natural order as well as matrix order. We can add A and B.

20 Neutrosophic Super Matrices and Quasi Super Matrices

3I1 I I2 0 212 2 I  A + B = 20 2322012I0.  I151I522I615I2

We see A+B is also a neutrosophic super matrix of natural order 310 and matrix order 24.

It is important to mention that even if A and B are two neutrosophic super matrices of same natural order and same matrix order yet the sum of A and B may not be defined.

This is established by an example.

Example 2.17: Let

I0123I0  1I0321I aaa11 12 13  A = 01I2340 =   and aaa21 22 23  501I0I1 2340I1I

5I33I1I  01420I0 b11bb 12 13  B = I32121I =   b21bb 22 23  1233410 0I15I1I

be two super neutrosophic matrices of natural order 57 and matrix order 23. Clearly A+B, i.e., the sum of A with B is not defined as a11  b11, a12  b12, a21  b21 and a22  b22. Hence the claim.

Thus we can restate the definition of the sum of two neutrosophic super matrices as follows. Neutrosophic Super Matrices 21

DEFINITION 2.7: Let A and B be two super neutrosophic matrices of same natural order say mn and matrix order st. The addition of A with B (B with A) is defined if and only if the natural order of each aij equal to natural order of bij for every 1  i  s and 1 j  t.

aa bb 11 1t 11 1t i.e., A =  and B =    aas1st bbs1st where aij and bij are simple matrices or the component submatrices of A and B respectively; 1  i  s and 1  j  t.

On similar lines the subtraction or difference between two neutrosophic super matrices can be defined.

Another important fact about the addition of neutrosophic super matrices is that sum of two neutrosophic super matrices need not in general give way to a neutrosophic super matrix it can be only a super matrix.

We illustrate this by an example.

Example 2.18: Let

105I2  160 I1   1234I 0102 I A =  and B =   I0123 I21 1 0   67890  0121 2 be two super neutrosophic matrices of same natural and matrix order.

Since each of the corresponding component matrices are equal their sum is defined.

22 Neutrosophic Super Matrices and Quasi Super Matrices

26 5 0 3   13 3 6 0 Now A+B =   . 02 2 3 3   6 8 10 10 2

We see A+B is only a super matrix and is not a neutrosophic super matrix.

Now we can have the following type of super neutrosophic matrices.

DEFINITION 2.8: Let

V1    V V =  2       Vn 

be a type A super vector where each Vi’s are column subvectors; 1  i  n. If some of the Vi’s are neutrosophic column vectors then we call V to be a type A neutrosophic super vector or neutrosophic super vector of type A.

We illustrate this by some examples.

Example 2.19: Let

 3     I   0    V1   1    V = = V ,  1   2      V3   0     4     I  Neutrosophic Super Matrices 23

V is a neutrosophic super vector of type A.

Example 2.20: Let

 I     2   1    V1   0      V2  V =  1  = , V3   1      V4   I     1   I 

V is a again a neutrosophic super vector of type A. All super vectors of type A need not be neutrosophic super vector of type A.

We illustrate this by an example.

Example 2.21: Let

 1     2   3     4    V1  5   V =   = V  6   2      V3   9   8     0     1 

24 Neutrosophic Super Matrices and Quasi Super Matrices

be a super vector of type A. Clearly V is not a neutrosophic super vector of type A.

Example 2.22: Let

 3     0   1    4   V1    V =   = V  1   2     2  V3     9   8     7 

be a super vector of type A and is not a neutrosophic supervector.

V1  Let V = V2 be a super vector of type A.  V3

The transpose of V is again a super vector which is a row supervector given by

t  ttt V = [3 0 1 4 | 1 2 | 9 7 8] = VVV123 .

We now define neutrosophic super vector of type B.

DEFINITION 2.9: Let T = [a11 | a12 | … | a1n] be a neutrosophic super vector where aij is a mmi matrix 1  i  n. T is a neutrosophic super vector of type B. We call the transpose of this neutrosophic super vector of type B to be also a neutrosophic super vector of type B. Neutrosophic Super Matrices 25

We illustrate this by the following examples.

Example 2.23: Let 021   5I0 112   a1  I01   A = a = 013 ,  2      a3  0I0 103   114   I25

A is a neutrosophic super vector of type B.

Example 2.24: Let

21356   I0123 4I567   00I01 10023   0I450 A1    A = 70311 = A .    2    3011I A3    I6100 025I0   1I011 23612   3I013

A is a neutrosophic super vector of type B. 26 Neutrosophic Super Matrices and Quasi Super Matrices

We wish to say super vector of type B are in general not neutrosophic.

We illustrate this situation by an example.

Example 2.25: Let 315   021 311   402 A1    A = 116 = A ,    2    789 A3  123   456   789

A is only a supervector of type B which is not a neutrosophic super vector of type B.

We define the transpose of a neutrosophic supervector of type B.

DEFINITION 2.10: Let

 A1    A =      An  be a neutrosophic super vector of type B.

The transpose of A denoted by

t A1  A At = 2  AAtt A t   12 n  Ab Neutrosophic Super Matrices 27

here each Ai is a neutrosophic matrix which has same number of columns. A is also a neutrosophic super vector of type B.

We illustrate this by the following example.

Example 2.26: Let

30I   123 987   760   A1  894   A =   = A 140  2  A  271  3    390   52I   123 be a neutrosophic supervector of type B.

t A1 t   tt t A = AAAA2123   A3

3197812351   = 0286947922 I3704010I3

is again a neutrosophic super vector of type B.

28 Neutrosophic Super Matrices and Quasi Super Matrices

As in case of neutrosophic super matrices we can define the sum of two neutrosophic super vectors of type A and type B.

DEFINITION 2.11: Let

a1    A1    A =        a Am  n1 an

be a neutrosophic supervector of type A.

Let the natural order of A be n  1.

b1   B1    Suppose B =         Bm   bn

B is also a neutrosophic super vector of type A of same natural order say n  1.

The sum of A and B denoted by A+B is defined if and only if the natural order of Ai is equal to the natural order of Bi for every i, i = 1,2,…, m i.e., both the matrices A and B have same matrix order.

We illustrate this by the following example.

Neutrosophic Super Matrices 29

Example 2.27: Let

0 I   1 0 I A  1 B  1 1       A = 0 A2  and B = 1 B2  A  B  2  3  1  3  3 0   I 2

be two neutrosophic super matrices of type A.

0  I   I      1  0   0  I  1   I1      A + B = 0 +  1  =  1  .     2  1   3  3  0   3      I  2   2I 

Clearly A+B is also a neutrosophic super vector of type A same natural order and matrix order that of A and B.

We can define also the subtraction of neutrosophic super vector of type A. Now A–B is calculated.

30 Neutrosophic Super Matrices and Quasi Super Matrices

0  I   I      1  0   1  I  1   I1      A – B = 0 –  1  =  1  .     2  1   1  3  0   3      I  2   I2 

We see A–B is also a neutrosophic super vector of type A of same natural order and matrix order as that of A and B.

However we wish to state that sum of difference of two neutrosophic supervector of type A need not always give way to neutrosophic super vector of type A it can also be only a supervector of type A.

We illustrate this situation by an example.

Example 2.28: Let

 I   I       1   1   0   0       2   1      A =  1  and B =  4   I   I       0   2   2   1       3   0 

be two neutrosophic super vector of type A.

Neutrosophic Super Matrices 31

 I   I   0         1   1   2   0   0   0         2   1   3        A + B =  1  +  4  =  5   I   I   0         0   2   2   2   1   3         3   0   3  is only a super vector of type A and not a neutrosophic super vector of type A.

We just indicate by an example how the minor product of type A neutrosophic super vectors is carried out.

Example 2.29: Let

A = [3 I 0 | 1 2 1 4 | 0 I] [A1 A2 A3] and

0  1 I  0 B1    B = 1  B .   2    2 B3  0  2  0 32 Neutrosophic Super Matrices and Quasi Super Matrices

0   1 I   0 The minor product of AB = [3 I 0 | 1 2 1 4 | 0 I] 1   2 0   2   0

0 0    1 2 = [3 I 0] 1 + [1 2 1 4] + [0 I]   2 0 I   0

= {30 + I1 + 0I} + {10 + 21 + 12 + 40} + {02 + I0} = {0 + I + 0} + {0 + 2 + 2 + 0} + [0 + 0] = I + 4 + 0 = I + 4.

Now in general minor product of A and B, two neutrosophic super vectors of type A may not be defined. If A is the neutrosophic super row vector of type A with A = [A1 … At] with natural order 1  n and matrix order 1  t then if B is a neutrosophic column super vector of type B then the minor t product of A with B = [B1… Bt] is defined if and only if natural order of B is n  1 and the matrix order of B is t  1 with natural order of each Bi in B is the same as the transpose of the natural order of each Ai in A for i = 1, 2, …, t or equivalently the natural order of Ai in A is equal to the order of the transpose of the natural order of Bi in B for each i = 1, 2, …, t.

Neutrosophic Super Matrices 33

Thus we see the minor product of a neutrosophic super row vector A with the transpose of the A is well defined for all the conditions required for the compatibility of the minor product of neutrosophic super vectors of type A are satisfied.

We will illustrate this situation by some examples before we proceed onto define more types of products on these neutrosophic super vectors of different types.

Example 2.30: Let

A = [1 0 I 2 | 0 0 1 | 2 3 I 0 1] be a neutrosophic super vector of type A of natural order 1  12 and matrix order 1  3.

 1     0   I     2     0  t  0  Clearly A =    1   2     3     I   0     1 

is also a neutrosophic super vector of type A for which AAt, the matrix minor product is well defined. 34 Neutrosophic Super Matrices and Quasi Super Matrices

 1     0   I     2     0  t  0  AA = [1 0 I 2 | 0 0 1 | 2 3 I 0 1]    1   2     3     I   0     1 

2 1    0 3 0     = [ 1 0 I 2]  + [0 0 1] 0 + [2 3 I 0 1] I I      1 0 2 1

= {11 + 00 + II+ 22} + {00+ 00 + 11} + {22 + 33 + II + 00 + 11}

= {1+0+I+4} + {0+0+1} + {4+9+I+0+1}

= {5+I} + {1} + {14 + I}

= 20 + 2I.

Thus we have the following theorems the proof of which is left as an exercise for the reader.

Neutrosophic Super Matrices 35

THEOREM 2.1: Let A = [A1 A2 … At] be a neutrosophic super row vector of natural order 1 n and matrix order 1 t. Then AAt the minor product of AAt exists (well defined).

THEOREM 2.2: Let A be a neutrosophic super row vector of type A. If B denotes the matrix A without partition then AAt = BBt.

Now for the same type A neutrosophic super vector A we define the major product.

DEFINITION 2.12: Let

A1    A =    and B = [B1 … Bs]    At 

be two neutrosophic super vectors of type A.

The major product of A with B is defined by

A1    AB =    [B1 … Bs]    At 

 2111  1BABABA s     BABABA =  2212 2 s  .        stst  BABABA st 

Here each Ai and Bj are matrices for 1  i  t and 1  j  s.

We illustrate this situation by the following example.

36 Neutrosophic Super Matrices and Quasi Super Matrices

Example 2.31: Let

3  2 I  A = 2 and B = [2 6 0 | 1 I]  0 I  4

be two neutrosophic super vectors of type A.

 3     2   I    AB =  2  [2 6 0 | 1 I]    0   I     4 

33    2260  21I II   22  = 260  1I 00  II  260  1I 44   

Neutrosophic Super Matrices 37

 618033I    412022I 2I 6I 0 I I    =  412022I .  00000   2I 6I 0 I I     824044I

Example 2.32: Let

 0     1   I     2   3  A =   and  4     1   I     2     3 

B = [0 1 | I 2 0 | 3 4 0 | I 0] be any two neutrosophic super vectors of type A.

We obtain the major product of the super column vector A with the super row vector B of the two neutrosophic matrices.

38 Neutrosophic Super Matrices and Quasi Super Matrices

0  1 I  2 3 AB =  [0 1 | I 2 0 | 3 4 0 | I 0] 4  1 I  2  3

00   0  0     101  1I20    1340    1I0      II   I  I 22   2  2     33 3 3 = 01   I20    340    I0 44   4  4     11   1  1  II   I  I     2012I2023402I0          33   0  0     

Neutrosophic Super Matrices 39

000 000 0000  01 I 2 0 3 4 0 I 0 0I I 2I03I4I0 I 0  022I4 0 6 8 02I0 0 3 3I 6 0 9 12 0 3I 0 = . 0 4 4I 8 0 12 16 0 4I 0 01 I 2 0 3 4 0 I 0  0I I 2I03I4I0 I 0  022I4 0 6 8 02I0 033I600 0000

Major product of two neutrosophic super vectors of type A is illustrated below by the following example.

Example 2.33: Let

 2     3   I     1    A =  2  and B = [3 0 1 | 0 I 1]  0     0   1     1 

be any two neutrosophic super vectors of type A.

40 Neutrosophic Super Matrices and Quasi Super Matrices

210   3013  3012 3011 I01 BA =  210   0I13  0I12 0I11 I01

32031I310210300111   =  02I31I 01I210  00I111 

60I 300  001  =  03II 02I0  0I1 

6I 3 1 =   .  4I 2I 1 I

Now we proceed onto define the minor product of neutrosophic super vectors of type B.

We only illustrate this by numerical examples for the reader to understand the minor product without any confusion as it will involve lot of notations and symbols repelling non mathematicians away from using these matrices.

Example 2.34: 01   I2 21301I   10 Let A = 000100 and B =   be any two 31 I0I001 10   21 neutrosophic super vectors of type B.

Neutrosophic Super Matrices 41

The minor product of AB is defined as follows:

01   I2 21301I     10 AB = 000100   31 I0I001 10   21

21  3 01I  31 01   = 00  010 10010  I2 I0  I 00121 

I4 30  12II      = 00 00  3 1  0I I0  2 1 

43I4I     =  31 .  2I 1I 

We from the numerical illustration observe the following facts.

1. The minor products of neutrosophic super vectors of type B is always a simple matrix. It may be a simple neutrosophic matrix or may not be a simple neutrosophic matrix.

B1    2. If A = [A1 … At] and B =    are neutrosophic   Bt  supervectors of type B, the minor product AB is defined if and only if 42 Neutrosophic Super Matrices and Quasi Super Matrices

(a) the number of submatrices in A and B are equal.

(b) The product AiBi is compatible, i.e., defined for each i, 1  i  t.

(c) The resultant simple matrix is of order mn where m is the number of rows of Ai and n is the number of columns of Bi which is the same for each Ai and each Bi i.e., number of rows of A is the same as that of Ai and the number of columns of B is the same as that of each Bi.

We give yet another numerical illustration to make one understand the working.

Example 2.35:

00I110210  Let A = 1011I0012 and I10001201

201I3   11000 00I10   12011 B = 0123I   12001 10I10   20101   31011

be two neutrosophic super vectors of type B.

We now find the minor product of A with B.

Neutrosophic Super Matrices 43

201I3   11000 00I10   00I11021012011  AB = 1011I00120123I   I1000120112001 10I10   20101   31011

201I3 00I1 10 11000  0123I 1011 I0  00I10 12001 I100 01 12011

21010I10      01220101  20131011 

12I1I10123I   = 321II240I2I3II  2I 1 1 I I 3I 1 2 0 0 1

402I121     82 1 2 3 512I3 1

 533I32I6I2   =  114I3I24I4  I7 . 2I 7 4 3I I 3 3I 2

44 Neutrosophic Super Matrices and Quasi Super Matrices

Thus we see if A is a neutrosophic super vector of type B. the transpose of A is denoted by At then AAt is just neutrosophic simple matrix or a simple matrix.

We will illustrate this by an example.

Example 2.36: Let

512411234  01011I123 A = 103200I00  200I01010 I15010101

be a neutrosophic super vector of type B. We now find the minor product of AAt.

5012I   11001 51241123420305   01011I123412I0 AAt = 103200I00 11001   200I01010 1I010 I1501010121I01   32010   43001

512 41   0105012I 11  412 I 0   = 10311001 20   11001 20020305 I0 I15 01 Neutrosophic Super Matrices 45

1234 1I010 I123  21I01  0I00  32010 1010  43001 0101

30 1 11 10 5I 11 17 5 8 4I 1  1100 1 522I1 = 11 0 10 2 I 15  8 2 4 2I 0  0 0 2 4 2I 4I I 2I I 0 5I 11 1 I 15 2I I  26 1 1 0 0 1

30 20 I 2I 4 614 I  I2014I I I2  4  2I I I 0 I  4I202 0 64I02

77 26 I 19 2I 14  4I 5I  18  I2617I 2I 2I2 6 = 19 2I 2 I 14 I 2 2I 2I 15 .  14 4I 2I 2 2 2I 6 I 2I 5I 18 6 2I 15 2I I 29

We see AAt is a symmetric simple neutrosophic matrix. It is always a square simple matrix. This simple matrix may be neutrosophic or may not be neutrosophic but it is always symmetric.

We give yet another numerical illustration. 46 Neutrosophic Super Matrices and Quasi Super Matrices

Example 2.37:

Let A be a neutrosophic supervector of type B given by

410213I10521012  2010120010I0600 A =  53I301210101I10  1I010300I001001

410213I10521012   2010120010I0600 AAt =   53I301210101I10   1I010300I001001

4251  103I 01I0  2031 1100  3213 I020  1010  010I 5010  2I00 1011  06I0  1010 2001

Neutrosophic Super Matrices 47

41 0   204251  1 =  01I0 53103I  I   1I 0

213I 2031     01201100     30123213     1030 I020 

1010 10521 012 010I 06I0 010I0 600  5010 1010 10101 I10  2I00 2001 0I001 001 1011

 178 234I  0000     84102 01I0 =     23 10 34 5 3I 0 I I 0     4I 2 53I1I 0000 

 17 8 9 2I 11   752 6    92I2 146    11 6 6 10

31 2I 7 1 5 0 1 2    2I 1 I 0 I 0 36 6I 0     7031 16I1I0      1I11I2001   48 Neutrosophic Super Matrices and Quasi Super Matrices

67I 152I402I18I    15 2I 11 I 12 7I 8 I = . 40 2I 12 7I 52 2I 12 3I  18 I 8 I 12  3I 13  2I

AAt is clearly a neutrosophic square symmetric matrix of order 4.

Now we proceed on to illustrate major product of neutrosophic super vectors of type B by an example.

Example 2.38: Let

310  121 I01  10130121 012   A =  and B = 21011002 60I    0I10010I 102 310  I00

be two neutrosophic super vectors of type B. To find

310  121 I01 10130121 012  AB =  21011002 60I  0I10010I 102 310  I00 Neutrosophic Super Matrices 49

10  1  30121  310  310  310   21   0  11002  121  121 121 0I  1  0010I   10  1  30121      I0121  I010  I0111002    0I  1  0010I   012 012 012   60I10 60I1 60I 30121      10221 1020 102 11002    3100I 31010010I 310   I00 I00 I00  

51 310136 5   52I 2 52 2 2 4I III13I0I12I2I    212I2 1 1 2 0 22I =   . 6 I 6 I 18 0 6 I 12 6 I    12I3 303 212I  51 310136 5   I 0 I3I0I2II

AB is clearly a neutrosophic super matrix. It may be a super matrix not necessarily neutrosophic.

Now we illustrate the major product of neutrosophic super vector of type B by an example.

50 Neutrosophic Super Matrices and Quasi Super Matrices

Example 2.39:

Consider the two neutrosophic super vectors.

2101  0110 1I00 10010010I I010   0I000I000 A = 001I and B =   . 001000010 1I00   0000I0001 000I  1000  0001

The major product of the neutrosophic super vector A with the neutrosophic super vector B is described in the following:

2101  0110 1I00 10010010I I010  0I000I000 AB = 001I  001000010 1I00  0000I0001 000I  1000  0001 Neutrosophic Super Matrices 51

100  10  010I  2101  2101  2101   0I0   00 I000  0110 0110 0110 001000010     0I00  0I00  0I00   000  0I 0001   100  10010I    I0100I0  I01000 I010I000   001I 001  001I 00001I 0010    000 0I0001 1I001001I00101I00010I  000I 0I0 000I 00 000I I000 1000001 100000 10000010    0001000 00010I 00010001 

2I02II202I1  0I100I01 0 0I000I00 0  I01I00I1 I = 0010I001 I .  1I010I10 I 0000I000 I  10010010 I  0000I000 1

We see AB is also a neutrosophic super matrix.

Now we illustrate by examples the major product of a neutrosophic super vector of type B with its transpose.

52 Neutrosophic Super Matrices and Quasi Super Matrices

Example 2.40: Let

301   0I0 100   100 00I A =   001 100   010   I01 0I0

be a neutrosophic super vector of type B.

Now the transpose of A denoted by

30110010I0 t  A = 0I0000010I 1000I10010

301  0I0 100  100 30110010I0 00I   AAt =  0I0000010I= 001   1000I10010 100  010  I01 0I0 Neutrosophic Super Matrices 53

301301 30110 301010I0  0I00I0 0I000 0I00010I 100100 1000I 10010010  30110010I0    100  100   100  0I0000010I     00I  00I  00I 100  0I 10010  001  001  001    10030110010I0  100   100       0100I0000010I  010  010       I01100  I01  0I  I01  10010  0I0  0I0  0I0    

10 0 3 3 I 1 3 0 3I 1 0  0 I000001 0 I 3 0110000 I 0  30110010I0 I000II00I0 = . 1000I10010 30110010I0  0 I000001 0 I  3I10III1I0I10 0 I00000I 0 I

We see in the first place AAT is also a neutrosophic super matrix. Further AAt happens to be a symmetric neutrosophic super matrix.

Thus as in case of usual matrices we get in case of neutrosophic row super vector A of type B, AAT happens to be a symmetric neutrosophic super matrix.

54 Neutrosophic Super Matrices and Quasi Super Matrices

We illustrate this yet by an example as we are more interested in the reader to apply and use it, that is why we are avoiding the notational difficult way of expressing it as a definition.

Example 2.41: Let

01010010  00I01101 A =  000110I0  I0000110

be a neutrosophic super vector of type B.

000I   1000 0I00   1010 At =   . 0110   0101 10I1   0100

Neutrosophic Super Matrices 55

Now we find

000I  1000 0I0001010010   1010 00I01101 AtA =    0110 000110I0   0101 I0000110 10I1  0100

000I 000I  01010 010 1000 1000 00I01 101 0I00  0I00  00011 0I0 1010  1010  I0000110 0110 0110      01010 010 0101 0101   00I01  101  10I110I1  00011  0I0 0100 0100   I0000 110    

I00 0 0I I 0  010 1 00 1 0 00I 0 I I 0 I  010 2 101I0 = . 00I 1 21 I 1  I0I 0 12 1 1 I101II12I0  00I 0 11 0 1

We see clearly the neutrosophic super matrix is symmetric. 56 Neutrosophic Super Matrices and Quasi Super Matrices

We get yet another example.

Example 2.42:

30110   0001I 10001   0I000 Let B =   2000I   00100 01100   10010

be a neutrosophic super vector of type B.

30102001   000I0010 Bt = 10000110   11000001 0I10I000

Consider

30110  0001I 30102001 10001   000I0010 t 0I000 BB = 10000110 2000I   11000001 00100 0I10I000 01100  10010 Neutrosophic Super Matrices 57

 3010    3011000030110 I   0001I1000001I 0   10001110100010  0I10     3010     000I  0I0001000I0000     1100  0I10     3010      000 I 2000I 2000I 100 0 00100 00100   1100     0I10  3010     000I 01100 01100 1000 10010 10010   1100     0I10

58 Neutrosophic Super Matrices and Quasi Super Matrices

2001   3011000 3011010  0001I01 0001I 10   1000100 1000101 I000   2001   0010 0I00001 0I00010   0001 I000   20 01    00 10 2000I 2000I  01 10 00100  00100 00 01    I0 00 2001   0010 01100 01100  0110 10010  10010  0001   I000

11 1 3 0 6 1 1 4  11I I 0 I 001 3 I 2 02I001  00 0I00I0 = . 6I2I04I002  10 000110 10 0I0120  41 102002 Neutrosophic Super Matrices 59

It is easily verified BBT is again a neutrosophic super matrix which is symmetric.

Example 2.43:

20  0I 00 0110101 Let A =  and B =   10 0000000 11  01 be a neutrosophic super vectors of type B.

20   0I 00 0110101 Now AB =     10 0000000 11   01

2001 2010101    0I 00 0I 00000  00  00 =   1001   1010101     1100   1100000    01 01   60 Neutrosophic Super Matrices and Quasi Super Matrices

0220202  0000000 0000000 = . 0110101 0110101  0000000

We see AB is a super matrix. Clearly AB is not a neutrosophic super matrix.

Thus it is pertinent to mention here that the major product of two neutrosophic super vectors of type B need not in general be a neutrosophic super matrix; it can only be a super matrix.

This is seen from example.

Now we proceed onto show by a numerical example how the minor product of two super neutrosophic matrices which has a compatible multiplication is carried out.

Example 2.44:

Let A and B be any two super neutrosophic matrices, where

10I010 1001000I0 21010I   010001010 I10001   100II0I00 A = 011100B =   100001100 001010   010100010 0000I0   001010101 301101

Neutrosophic Super Matrices 61

10I010 1001000I0 21010I   010001010 I10001  100II0I00 AB = 011100   100001100 001010  010100010 0000I0  001010101 301101

 1     2   I      0  [1 0 0 1 | 0 0 | 0 I 0] +    0   0     3 

0I  10 10 010001010 11  100II0I00 01   00  01 62 Neutrosophic Super Matrices and Quasi Super Matrices

010  10I 001100001100    100010100010 010001010101   0I0  101

111  1001 00   0I0  222  III     000  = 1001 00  0I0  000     000   31 0 0 1 3 0 0   3 0 I 0   

0I0100  0I01 0I010   10100I  10 I0 10 I00  10 10  10    110100  1101  11010 +  01100I  01 I0  01 I00    00 00 00    0100 01 010 0101  01  100I I0 I00 

Neutrosophic Super Matrices 63

 1000  01 100 010 010   010  0101  00  010 10I10I  10I       0010  10 101 001 001  001  1000   01 100  100100  100   0101  00  010 010 010   010  0010   10 101 0I0 0I0  0I0     1000  01 100    1010101 10100 101010       0010  10 101   

1001000 I 0 I00I I0I00  20020002I0 010001010 I00I000 I 0 010001010  0000000 0 0 110I I1I10 0000000 0 0 100I I0I00  0000000 0 0 000000000  30030003I0 100I I0I00

010100 0 10  10I0I11I0I 001010 1 01   100001 1 00 010100 0 10  0I0I00 0 I0  101011 2 01 64 Neutrosophic Super Matrices and Quasi Super Matrices

1I102I I 0 I 1I  0  31I2 I21I2I1I I11I 111 1I1  = 210I I21I1 0. 1101II0 I 1 0  0I0I 000 I 0  5013I1I12I3I1 

Thus the minor product is again a super neutrosophic matrix.

We give yet another illustration.

Example 2.45:

Let A and B be two neutrosophic super matrices where

1011200  0I0010I I121010  A = 00I0101 and 1I01I00  00100I0  11201I1

011I00I   0110101 I001000   B = 10I1001 0100100   1100I0I   0000011 Neutrosophic Super Matrices 65

1011200011I00I  0I0010I  0110101 I121010I001000   AB = 00I010110I1001 1I01I000100100   00100I01100I0I   11201I10000011

101     0I0  I12    0110101 = 00011I00I I   I001000 1I0      001     112 

1200  010I 10I1001 1010  0100100 + 0101  1100I0I 1I00  0000011 00I0  01I1

66 Neutrosophic Super Matrices and Quasi Super Matrices

Here the usual product of each submatrices are performed and we see in this minor product of neutrosophic super matrices each cell is compatible with usual product of submatrices.

11   1 01   1I00   1 00   0 II   I    00   0 = 01  1I00   1 + 11   1    00   0  10 1   11 I 0 0   1  

0101 011010  011   I0I0 I00100  I00   12  12 12     0I01  0I1010    0I1       + I0I0  I00100    I00     01 01 01    01  1010   1 12  12  12  I0 0100 0     

Neutrosophic Super Matrices 67

10 I100  1     120001 12000010    12000    010I 11 010I  00I0   010I  I     00 0001  1  101010 1010I10010101     010101 01010010 01010    1I0011 1I0000I0 1I00I     00I000 00I0000100I01  10 I100  1     01 0010 0 01I1  01I1   01I1  11 00I0  I     00 0001 1    

011I00I I 001000    0000000 0 I I0I0I  0III00I 2I112101    = 0000000  I 00I000  011I00I 0 II0I0I    0000000 I 001000    011I00I 2I112101 

12I1201   01001I I 21I1I01I     0100111 1II1I01   II00I0I   I1I001I11I 

68 Neutrosophic Super Matrices and Quasi Super Matrices

1I 3 1I 2I 2 0 1I  01II 01II2I 22II2 2I1I31I022I    = I10I111. 1 1 2I 1 2I I 1 2I 0 2I  1  2I I 0 1 I 0 I  3I 3 I 2 2 I 2 I 1 2 2I

We see AB is again a neutrosophic super matrix under the minor product of neutrosophic matrices.

Now we show the minor product of a neutrosophic super matrix with its special transpose can be defined.

Let us first illustrate by a few examples the special transpose of a super neutrosophic matrix.

Example 2.46:

21312 I 0   1I01011 01I1102 Let A =   3I10101 0101010   I110I00

be a neutrosophic super matrix.

Neutrosophic Super Matrices 69

Now the special transpose of A denoted by At is given by a neutrosophic matrix.

21030I   1I1I11 30I101 t   A = 111010. 20110I   I10010   012100

Example 2.47:

21000303210  0152I I1012I 7890712I012  0I010I60120 Let A = 10I01211230  001001I I127 10011021023  0I10I1I0I12  10102311101 be a super neutrosophic matrix.

To find the transpose of the neutrosophic super matrix A we take the transpose of each of the submatrices in A. 70 Neutrosophic Super Matrices and Quasi Super Matrices

207010101  118I000I0 0590I1011  020100100 0I70101I2 t  A = 3I1I21013 ; 01261I2I1  30I01I101  2101210I1 121232210  0I2007321

clearly At is also a neutrosophic super matrix.

At will be known as the special transpose of A.

Clearly if A is a neutrosophic row super vector of type B then the special transpose of A, At is a neutrosophic column super vector of type B.

Similarly if A is a neutrosophic column super vector of type B the At the special transpose of A is a neutrosophic row super vector of type B.

We illustrate this by a few examples.

Example 2.48:

Let A = [2 1 0 | 1 1 1 I 0 | 0 3 I 0] be a neutrosophic super row vector then

Neutrosophic Super Matrices 71

2  1 0  1 1  1 At = , At is a neutrosophic super column vector. I  0  0 3  I 0

Example 2.49:

3  6I 1  2 0  Let A = 1 a neutrosophic super column vector. 1  I  0 2  I

Then the special transpose of A; At = [3 6 I 1 2 0 | 1 1 I 0 | 2 I ]. Clearly At is a neutrosophic super row vector. 72 Neutrosophic Super Matrices and Quasi Super Matrices

Example 2.50:

310 1 I 3 12 I  121040123I Let A =  I0I 234I201  0143I0 1 10 1

be a neutrosophic super row vector of type B.

Then At the special transpose of A.

t  tt t That is if A = [A1 A2 A3] we see A = AAA123 and the row vectors are arranged as column vectors so that the number of columns of A is equal to the number of rows of At.

31 I 0   1201 01 I 4   10 23I At = I430   304I1 11 2 1   22 0 0   I3I1 1

is neutrosophic super column vector of type B.

Neutrosophic Super Matrices 73

Example 2.51: Let

 0I120    100I0  1010I    2 345 0  010I2    10015I A = 7I 0 2 1 3    01011    11111  IIII3    20410  5123I

be a neutrosophic super column vector of type B. The special transpose of A is

01120 1 7I01I  25  I0031 0 0 11I 0 1 At = 10140 0 2 01I 4 2 .  2I05I 1 1 11I 1 3 00I025I3 113 0 I

We see At is a neutrosophic super row vector of type B.

We now illustrate the minor product of the neutrosophic super matrices A with its transpose.

74 Neutrosophic Super Matrices and Quasi Super Matrices

Example 2.52:

3011001201  0I00110010 101I00I101  41021I0012 Let A =  0103000I01  0111210201 I000101010  12010I00I0

be a neutrosophic super matrix.

To find the minor product of A with At.

301400I1   0I011102 10100100   10I23101 01010210 At =   010I010I 10I00010   2010I200   0101001I 10121100

Neutrosophic Super Matrices 75

3011001201   0I00110010 101I00I101   41021I0012 AAt =   0103000I01   0111210201 I000101010   12010I00I0

301400I1   0I011102 10100100   10I23101 01010210   = 010I010I 10I00010   2010I200   0101001I 10121100

301 10   0I0 01 101 I0 301400I1  4102110I23101  0I011102   010 3001010210  10100100  011 12 I00 01   120 10

76 Neutrosophic Super Matrices and Quasi Super Matrices

01201  10010 010I010I 0I101  10I00010 I0012 + 2010I200 00I01  0101001I 10201 10121100 01010  I00I0

301 301 301 3014   0 0I1 0I00I0  0I0 0I01  1 102 101 101  101 1010   0 100 410 410 410  3014  0 0I1    0100 I 01  0101 010102   1010  0 100  0113014   0110    0110I1    I000I01 I001 I00102  1201010 1200 120100  

Neutrosophic Super Matrices 77

10 10  10    0110I2  013  01101  I00101  I00  I0210    21 21  21  10I2 3 101  3030  30  01010210  12 12  12 10I2  3  101 01 01  01   0101 0 210 10 10  10  

010I 0 10I 01201 01201  01201 10I0  0 010 10010 10010 10010 2010  I 200 0I101 0I101  0I101 0101  0 01I I0012 I0012  I0012 1012 1 100 010I 010I   10I0 0010  00I012010I200I01  00I01 00   0101 0 01I   1012 1 100 010I 0 10I     1020110I0 10201 0  10201010   010102010 01010 I 01010200     I00I00101 I00I0 0 I00I001I     1012 1 100 78 Neutrosophic Super Matrices and Quasi Super Matrices

10 0 4 12 0 1 3I 3 1 0 I 2 3 1 0 1  0 I0I II02I0101 0210 402401I1 I0I2I3II0I  12 I 4 17 1 1 4I 6 2 1 2I 5 6 4 1 2  0 I 0 1 11 0 2 303I6 9 303  1I111202 12I43521 3I 0 I 4I 0 0 I I 0 1 0 1 0 2 1 0  32I1622I 5 10I 23101

603I22I1510  0201I0112I I3 0 2I 2 1I 3 I 0  21I2I52 I212I    2I1 0 I1 2 1I  2I10 0  513I22I160I 11I10 02I  02I02I0 II2I

17 0 7 2I 16 4 2I 7 3I  1 4  03I022II3I2  4I 7 2I 0 4 2I 6 2I 4I 1 4 I 2I 1 I  16 2 2I 6 2I I 27 9 I  7 4I  2 8 2I = . 42II 4I1911I52I0 5  73I4II752I1323I  3I 1 2 2I 4I 2 0 2 3 I 2I  4 4I 1 I 8 2I 5 3  I 2I 6  2I Neutrosophic Super Matrices 79

It is clear that AAt is a neutrosophic super symmetric matrix.

Thus minor product of a matrix A with its special transpose At gives a super symmetric matrix.

It is pertinent to observe A is not a symmetric neutrosophic matrix.

Hence At is also not a neutrosophic symmetric super matrix how ever the product is a symmetric super neutrosophic super matrix.

We give yet another example.

Example 2.53:

30I04000101  0100010010I 60000010001  71011000011 10010101100  00110010110 Let A = 1010I00I000  01I00I00I00  400010I0001 100I0011000  02110I00100 1000I010001

be a super matrix.

To find the minor product of A with 80 Neutrosophic Super Matrices and Quasi Super Matrices

306710104101  010100010020 I000011I0010  000111000I10 400100I0100I t  A = 0100100I00I0 00100100I101  000010I00100  1100110I0010 000101000000  1I1100001001

product of A with At.

30I04000101  0100010010I 60000010001  71011000011 10010101100  00110010110 AAt = 1010I00I000  01I00I00I00  400010I0001 100I0011000  02110I00100 1000I010001 Neutrosophic Super Matrices 81

306710104101  010100010020 I000011I0010  000111000I10 400100I0100I  0100100I00I0 00100100I101  000010I00100  1100110I0010 000101000000  1I1100001001

301  010 600  710 100 306710104101 001 010100010020 101 I000011I0010 01I  400 100  021 100

82 Neutrosophic Super Matrices and Quasi Super Matrices

0400  0010 0001  1100 1010000111000I10  1001400100I0100I  0I000100100I00I0  00I000100100I101  010I I001  10I0 0I01

0101  010I 0001  0011 1100000010I00100  01101100110I0010   I000000101000000  0I001I1100001000  0001 1000  0100 0000

Neutrosophic Super Matrices 83

  301306 30171010 3014101  010010 01010001 0100020 600100 6000011I 6000010  710  710 710    100306  10071010100 4101    001010  00110001001 0020        101100  1010011I101 0010 01I  01I01I  400  400 400 306   71010   4101 100 100  100  010  10001 0020 021  021   021  100   0011I  0010 100  100 100

000  11100   0I10  0400  0400   0400   400   100I0    100I  0010 0010 0010 010   0100I    00I0  0001  0001   0001   001  00100   I101   1100 1100 1100 00011100  0I10 10101010  1010   400 100I0  100I 10011001   1001   010  0100I    00I0  0I000I00   0I00   00100100 I101 00I0 00I0 00I0  010I 000 010I11100   010I  0I10        I001400 I001100I0   I001100I   10I0010 10I00100I   10I000I0         0I01001 0I0100100 0I01I101      

84 Neutrosophic Super Matrices and Quasi Super Matrices

000  010I0   0100  0101  0101   0101   110   0110I    0010  010I 010I 010I 000   10100    0000  0001  0001   0001   1I1  10000   1000   0011 0011 0011 000010I0  0100 11001100  1100   110 0110I  0010 01100110   0110   000  10100    0000  I000I000   I000   1I110000 1000 0I00 0I00 0I00  0001000 0001010I0 00010100  1000110 10000110I 10000010 0100000 010010100 01000000  00001I1 000010000 00001000 

10 0 18 21 3 1 4 I 12 3 1 3  0 1 0 1 000 1 0 0 2 0 18036426060 2460 6  2114250707 1 287 2 7 306 7101 0 41 0 1  100 0011 I 00 1 0 = + 406 7112 I 41 1 1  I10 10II1I002I0  1202428404 0 164 0 4  306 7101 0 41 0 1    12020112I0050   306 7101 0 41 0 1

Neutrosophic Super Matrices 85

160040 04I04 0 04I  0 100 1 0 0 I 0 0 I 0 0010 0 1 00 I 1 0 1  4002 1 1 I0 1 I 1 I 0101 2 1 0I 0 I 1I0  0011 1 2 00 I 1I 1 1 + 4I 0 0 I 0 0 I 0 I 0 0 I  0I00 I 0 0I 0 0 I 0  40I1 0 I I01I I 0 2I 001I I 1I00 I 1I I 1    0I011I1 0I 0 I 2I0 4I 0 1 I 0 1 I 0 2I 1 0 2I

2 1I11110I1010  1I 2II I 110 I I 010 1 I 1100001000  1 I 1201001000 1 1 0021I I0110  1 1 01120I0010 = 0000I0I00I00  II00II0I00I0  1 I 1100001000 0 0 0010I00100  1 1 00110I0010 0 0 0000000000

86 Neutrosophic Super Matrices and Quasi Super Matrices

28 1 I 19 26 4 2 4 4I 2I 17 3 2 3  4I  1I 22II 1I  2 1 0 2I1 I 0 3I  0 19 I 38 43 6 1 6 0 25 I 7 0 7  26 1 I 43 54 8 2 7 I 1 30 7 I 3 7 I 4268521I2I42I2I1  21122512II1I31 44I0 6 7I1I 1 22II 4I1I 131I  2I 2I 1 0 1 2I 2I I 1 3I 0 0 2 3I 0  17 I 25 I 30 4 I 4 I 0 18 I 4 I 042I 3077I2I1I1I04I3II2    23I0 32I31323I0 I62I0 34I0 7 7I 1 1 1I 0 42I2 0 12I

We see the minor product of a neutrosophic super matrix A with its transpose At is a symmetric neutrosophic super matrix.

The entries in these super matrices can be replaced from the dual number neutrosophic rings, special dual like number neutrosophic rings, special quasi dual number neutrosophic rings and special mixed dual number neutrosophic rings.

Chapter Three

SUPER BIMATRICES AND THEIR GENERALIZATION

In this chapter we introduce the notion of super bimatrices or equivalently will be known as bisuper matrices. The study of bimatrices has been carried out in [6]. Here we define super bimatrices and generalize them.

This chapter has two sections. Section one defines super bimatrices and enumerates a few of their properties. In section two the generalization of super bimatrices (bisupermatrices) to super n-matrices (n-super matrices) is carried out (n > 2) when n = 2 this structure will be known as bisupermatrices or superbimatrices. When n = 3 we call it as trisupermatrix or supertrimatrix. This can be made into a neutrosophic superbimatrix by taking the entries from Z  I or Q  I or R  I or C  I or Zn  I.

3.1 Super bimatrices and their properties

In this section we introduce the new notion of super bimatrices (bisuper matrices) and enumerate a few of its algebraic properties and operations on them. 88 Neutrosophic Super Matrices and Quasi Super Matrices

DEFINITION 3.1.1 : Let A = A1  A2 where A1 and A2 are super row vectors of type A, A1 = (a1, a2, …, ar | ar+1… | … | am … an) and A2 = (b1, …, bs | bs+1… | … | bt … br). Then we call A = A1  A2 to be a super birow vectors (bisuper row vectors or super row bivectors) of type A if and only if A1 and A2 are distinct in atleast one of the partitions.

We illustrate this situation by some examples.

Example 3.1.1: Let A = A1  A2 = (3 0 1 | 1 1 0 4 | –5 | 7 8 3 1 2)  (1 0 | 3 1 2 | 1 1 0 4| 6 1 0). A is a super row bivector (bisuper row vector) of type A.

Example 3.1.2: Let A = A1  A2 = ( 0 0 | 0 0 0 | … | 0 0 0)  (0 0 0 0 0 | 0 0 0 0 0| 0 0), A is a super row bivector of type A also known as the super row zero bivector of type A.

Example 3.1.3: Let A = A1  A2 = (1 1 1 1 | 1 1 1 1 1 1| 1 1)  (1 1 | 1 1 1 1 | 1 1 1 1 1 1 1) ; A is a super row bivector of type A. Infact A is also know as the super row unit bivector of type A or bisuper row unit vector of type A.

Example 3.1.4 : Let A = A1  A2 = ( 3 1 0 | 4 5 | 1 1 1 2 5)  (3 0 1 | 4 5 | 1 1 1 2 5); A is not a super row bivector for A1 and A2 are not distinct. If A = A1  A2 = (3 | 1 0 4 5 | 1 1 1 2 5)  (3 0 | 1 4 5 1 | 1 1 1 2 5) then A is a super row bivector of type A.

Now we proceed on to define the new notion of bilength of the super row bivector of type A.

DEFINITION 3.1.2: Let A = A1  A2 = (a1 … | …. | …| ar … an)  (b1 … | …. | …| bs … bn) be a super row bivector of type A where both A1 and A2 are of length n but they are distinct in the partitions. We say A is of bilength n and denote the bilength by (n, n). Super Bimatrices and their Generalization 89

If A = A1  A2 = (a1 … | …. | …| ar … an)  (b1 … | …. | …| bs … bn) be a super row bivector of type A (m  n) then we say A is of bilength (m, n).

We illustrate this by some examples.

Example 3.1.5: Let A = A1  A2 = (1 0 2 3 | 5 6 7 8 9 | 2 1)  (0 1 2 0 | 1 0 9 2 7 | 5 3) be a super birow vector of type A. We say A is of bilength (11, 11), i.e., this super birow vector has same bilength.

Example 3.1.6: Let

A = A1  A2 = (0 1 0 | 2 5 3 | 1 1 0)  (3 1 2 0 | 2 1 | 5) be a super birow vector of type A. A is of bilength (9,7). We see this super row bivector has different bilength for its component super row vectors A1 and A2.

We can define the notion of super row bivector equivalently in this form.

DEFINITION 3.1.3: Let A = A1  A2, if A1 and A2 are two distinct super row vectors of type A, then we call A to be a super row bivector of type A.

Now we proceed onto define the notion of super column bivector of type A (super bicolumn vector or bisuper column vector of type A).

DEFINITION 3.1.4: Let A = A1  A2 where A1 and A2 are super column vectors of type A which are distinct. Then we say A is a super column bivector of type A or bisuper column vector of type A or super bicolumn vector of type A.

We illustrate this situation by the following examples.

90 Neutrosophic Super Matrices and Quasi Super Matrices

Example 3.1.7: Let

 5  3    7 1    7 0     2  1 A = A1  A2 =   4  1     0  1  3  0     1  7    2 

be a super bicolumn vector of type A. Clearly A1 and A2 are distinct.

Example 3.1.8: Let

0 0    0 0    0 0    0 0 0 0    0 A = A1  A2 = 0  0 0    0 0    0 0    0 0    0 0 0

be a super column bivector of type A. We say A is the zero super column bivector of type A or super column zero vector of type A. Super Bimatrices and their Generalization 91

Example 3.1.9: Let 1  1 1    1 1    1 1   1 1  1 A = 1    = A1  A2. 1 1  1 1    1 1    1 1    1 1

A is a super column bivector of type A known as the super column unit bivector of type A.

DEFINITION 3.1.5: Let

 ab11       A = A1  A2 =         abmn

be a super column bivector of type A. If (m  n) then we say A is of super bilength (m, n).

If in A = A1  A2 where A1 and A2 are distinct super column vectors, if the length of A1 is the same as that of A2 then we say A is a super column bivector of length n and denote the bilength by (n, n). 92 Neutrosophic Super Matrices and Quasi Super Matrices

Example 3.1.10: Let

32   01 20   56 15   A = A1  A2 = 37 08   19   11 04   40

be a super column bivector of type A. A is of bilength (11,11), i.e., both A1 and A2 are of length 11. Clearly A1 and A2 are distinct super column vectors.

Example 3.1.11: Let 3  3 1    4 0     0  1    2  3   7  A = A1  A2 = 1     5  2   8  1     9  1    1    5    10 7

Super Bimatrices and their Generalization 93

be a super column bivector of type A. We say A is of bilength (11,10). Clearly A1 and A2 are of different lengths.

These new type of super birow vectors and super bicolumn vectors will be useful in computer networking, in coding / cryptography and in fuzzy models.

Now we proceed on to define the transpose of a super row bivector and super column bivector of type A.

DEFINITION 3.1.6: Let A = A1  A2 be a super row bivector of type A. We define the transpose of A as the transpose of each of the super row vectors A1 and A2, i.e.,

t t tt A = (A1  A2) =  AA 21 . Clearly the transpose of a super row bivector is a super column bivector of type A.

Likewise if A = A1  A2 is a super column bivector of type A then the transpose of A is a super row bivector of type A, i.e., t A is the union of the transpose of the column vectors A1 and A2 t t tt respectively. We denote A = (A1  A2) =  AA 21 .

We shall illustrate this by the following example.

Example 3.1.12: Let A = A1  A2

= (0 1 2 3 | 4 3 1 | 7 0 3 5 6 8)  (3 0 | 2 5 4 1 | 0 0 7 2 5) be a super row bivector of type A. 94 Neutrosophic Super Matrices and Quasi Super Matrices

0  13    20    32    45    34    The transpose of A denoted by At = 11 = AAtt .    12 70     00    37    52    65    8

It is clear that At is a super column bivector or super bicolumn vector of type A.

Example 3.1.13: Let A = 31   03 10   25 56   67 A  A = 18 1 2   09   33 71   42 55   94 Super Bimatrices and their Generalization 95

be a super column bivector of type A. The transpose of A t tt denoted by A = AA12 = [3 0 1 2 5 6 | 1 0 3 | 7 4 5 9]  [1 3 0 | 5 6 7 8 9 | 3 – 1 2 5 4].

It is easily verified that At is a super row bivector.

Now having seen the transpose when is the addition and multiplication of these super bivectors of type A is possible.

DEFINITION 3.1.7: Let A = A1  A2 and B = B1  B2 be two super row (column) bivectors of type A. We say A and B are similar or “identical in structure” (identical in structure does not mean they are identical) if the following conditions are satisfied.

1. The bilength of A and the bilength of B are the same i.e., if bilength of A = A1  A2 is (m,n) then the bilength of B = B1  B2 is also (m,n).

2. The partition of the row (column) vector Ai is identical with that of the row (column) vector Bi, true for i=1,2.

We illustrate this by the following examples.

Example 3.1.14: Let A = A1  A2

= [3 1 5 6 | 0 2 3 | 1 5 – 2 3 1 5]  [8 0 | 1 2 3 | 7 | 5 6 7 8]

be a super row bivector.

B = [0 4 0 1 | 6 – 12 | 3 0 1 1 0 7]  [3 5 | 0 1 0 | 8 | 3 0 1 5]

be another super row bivector. We can say A and B are identical in structure.

We see bilength of A is (13,10) and that of B is (13,10), they are equal.

96 Neutrosophic Super Matrices and Quasi Super Matrices

Example 3.1.15: Let A =

8 0 3   1    1  2 0   1    3  1 1   1    4  2 5 0 5 3 A1  A2 = 7    and B = 1    = B1  B2 .  7  4 8 2 1 5 1   3    2  9 8   4    0  8 0 5 3 7

Clearly the super column bivectors A and B are identical in structure and we see the bilength of A is (9,10) and that of B is (9,10) i.e., the bilength of A and B are the same.

Thus we see if two super bivectors of type A are identical in structure they also enjoy the property that they have same bilength. Only to make things clear we mention it as a separate property though the concept identical in structure covers the equal bilength.

However it is pertinent to mention here that equal bilength will not in general imply they are identical in structure. But identical in structure will always imply they have same bilength.

Now we proceed onto define the notion of addition of super bivectors of type A. When we say super bivectors of type A it implies the super bivector can be a sub birow vector of type A or super bicolumn vector of type A; i.e., the term super bivector of type A includes both row bivector as well as column bivector.

Super Bimatrices and their Generalization 97

DEFINITION 3.1.8: Let A = A1  A2 and B = B1  B2 be any two super row bivectors which are identical in structure. Then addition or subtraction of A with B denoted by

A + B = (A1  A2)  (B1  B2) is well defined and Ai  Bi is carried out component wise and A  B is again a super birow vector which are identical in structure with A and B.

On similar lines we can define the addition or subtraction of A with B in case of super column bivectors A and B which are identical in structure.

We illustrate this by the following examples.

Example 3.1.16: Let

3 4  2  0  0  1    1  1 1  1    0  2  2  2   3  3  A = 5  = A1  A2 and B = 3  = B1  B2  4 1 7  0   7  0  3 1  2  2  4  4    0 5 5 1

be two super column bivectors which are identical in structure. We find A+B the sum of A with B to be (A1 + B1)  (A2 + B2) 98 Neutrosophic Super Matrices and Quasi Super Matrices

7   2  1    0 0     2  4    6  = 8  = A+B.   3  7    7  4   4  8    5 4

Clearly A+B is again a super column bivector which are identical in structure with A and B.

Now A – B = A1 – B1  A2 – B2

1  2  1    2 2   2 0    0  = 2  .  5  7    7  2  0  0    5  6

We see A – B is again super row bivector of type A identical in structure with A and B. Thus we see the property of identical in structure is preserved under both addition and subtraction.

In view of this we have the following interesting result.

Super Bimatrices and their Generalization 99

THEOREM 3.1.1: Let S = {A = A1  A2 / the collection of all super row bivectors of type A which are identical in structures with A having its entries from Q or R or C or Zn or C(Zn) or 2 2 Z(g), Z(g1) or Z(g2) (g = 0 is a new element g1 = g1 is a new 2 element and g2 = –g2 a new element n is such that, 2  n  }. Then S is a group under the operation of addition.

The proof is left as an exercise for the reader.

THEOREM 3.1.2: Let B = {B = B1  B2 / the collection of all super column bivectors of type A which are identical in structure of type A with B having its entries from Q or R or C or 2 Zn or C(Zn) or Z(g) or Z(g1) or Z(g2) (g = 0 is a new element 2 2 g1 = g1 is a new element and g2 = –g2 a new element; n is such that, 2  n  }. Then B is a group under the operation of addition of the super column bivectors.

This proof is also simple left as an exercise for the reader.

Now we define the product of a super row bivector of type A with a super column bivector of type A where the multiplication is compatible.

We first show how the minor product of two super vectors are defined whenever the multiplication happens to be compatible.

This is illustrated by some examples.

Example 3.1.17:

Let x = [ 3 0 1 | 2 3 4 5] and y = [1 0 0 | 2 3 1 6]t.

The minor product of the super vectors x with y is given by

100 Neutrosophic Super Matrices and Quasi Super Matrices

1   0 0   xy = [3 0 1 | 2 3 4 5] 2 3   1   6

2 1     3 = [3 0 1] 0 + [2 3 4 5]     1 0   6  

= 3 + {4 + 9 + 4 + 30} = 3+47 = 50.

Example 3.1.18: Let  6    1  2     3  4    5   7  x = [0 1 2 3 | 4 5 | 7 8 9 0 1 2 3 | 8] and y =   8    9   6    1  2    3    8 Super Bimatrices and their Generalization 101

be two super vectors  6    1  2     3  4    5   7  xy = [0 1 2 3 | 4 5 | 7 8 9 0 1 2 3 | 8]   8    9   6    1  2    3    8

 7    8 6     9  1 4   = [0 1 2 3] + [4 5]   + [7 8 9 0 1 2 3]  6  + [8] [–8] 2  5   1 3    2    3

= {0–1 + 4+9} + {–16+25} + {49 – 64 + 81+0–1+4–9} + {–64}

= 12 + 9 + 60 – 64 = 17.

102 Neutrosophic Super Matrices and Quasi Super Matrices

DEFINITION 3.1.9: Let

W1    W x = [V V … V ] and y =  2  1 2 n      W  n 

be two super vectors of type A.

W1    W xy = [V V … V ]  2  1 2 n      Wn 

= V1 W1 + V2 W2 + … + Vn Wn.

Note:

The compatability of each Vi with Wi is well defined for i = 1, 2, …, n, this is illustrated by the following examples.

Example 3.1.19: Let

x = [1 2 0 1 5 | 3 2 0 | 7 0 – 1 2 3 4 8 | 8 0 1 4]

be a super row vector. Super Bimatrices and their Generalization 103

 1     2   0     1   5     3   2     0     7  x t   0    1  2     3     4   8     8   0     1   4 

1   2 3   xxt = [1 2 0 1 5] 0 + [ 3 2 0] 2 +     1 0 5 104 Neutrosophic Super Matrices and Quasi Super Matrices

 7    0   8 1     0 [7 0 –1 2 3 4 8] 2 + [8 0 1 4]     1  3      4  4     8 

= {1 + 4 + 0 + 1 + 25} + {9 + 4 + 0} + {49 + 0 + 1 + 4 + 9 + 16 + 64} + {64 + 0 + 1 + 16}

= {31 + 13 + 143 + 81}

= 268.

Thus we can define the product in case of super row vector x with xt which is as follows:

Vt   1  Vt If x = [V V … V ] then xt =  2  . 1 2 n     t  Vn 

Vt   1  Vt Now xxt = [V V … V ]  2  . 1 2 n     t  Vn 

tt t V11 V V 2 V 2 ... V n V n.

Super Bimatrices and their Generalization 105

We illustrate this by an example.

Example 3.1.20: Let

x = [0 1 | 3 4 0 1 | –3 2 1 6 7 8 9 | 3 2 1 4 5] then

 0     1   3     4   0     1  3    2    1 x t     6     7   8     9     3   2     1   4     5 

where xt is a transpose of x and x is a super row vector.

106 Neutrosophic Super Matrices and Quasi Super Matrices

 0     1   3     4   0     1  3    2    1 xxt  [01|3401| 3216789|32|145]    6     7   8     9     3   2     1   4     5 

 3   2 3      1  0 4   = [0 1]   + [3 4 0 1] + [–3 2 1 6 7 8 9]  6  1 0    7  1    8     9 

Super Bimatrices and their Generalization 107

3   2 + [3 2 1 4 5] 1   4 5  

= {0 + 1} + {9 + 16 + 0 + 1} + {9 + 4 + 1 + 36 + 49 + 64 + 81} + {9 + 4 + 1 + 16 + 25}

= 1 + 26 + 244 + 55

= 326.

Now we proceed onto define the notion of major product of two super vectors.

We shall only illustrate this by simple examples.

Example 3.1.21: Let

0  1 2  4 x = 5 and y = [1 2 | 0 3 4 5 | 8 9 2 1 0]  7 8  1  4 be two super vectors of type A.

The major product of x with y denoted by 108 Neutrosophic Super Matrices and Quasi Super Matrices

0   1 2   4 xy = 5 [1 2 | 0 3 4 5 | 8 9 2 1 0]   7 8   1   4

00   0    112  10345   189210   22   2  44   4    55 5 = 12   0345   89210  77   7    88   8  11   1 12   0345   89210  44   4  

0000 0 0 0 0 000  12034589210 2 4 0 6 8 1016184 20  4801216203236840  5 10 0 15 20 25 40 45 10 5 0  7 14 0 21 28 35 56 63 14 7 0 8 16 0 24 32 40 72 81 18 9 0  12034589210  4801216203236840 Super Bimatrices and their Generalization 109

Example 3.1.22: Let

1  4 3  7 0  2 5  1 x and y [340|21013|1214|1] 0 1  1 2  3 4  7  0

be two super vector.

The major product of x with y denoted xy =

110 Neutrosophic Super Matrices and Quasi Super Matrices

1  4 3  7 0  2 5  1 340|21013|1214|1 0 1  1 2  3 4  7  0

We see for major product to be defined we need not have

(1) The natural order of the super column vector to be equal to the transpose of the super row vector.

(2) The number of partitions of x need not be equal to the number of partitions of y.

Super Bimatrices and their Generalization 111

11   1 1 340  21013  1214  1 44   4 4 33   3 3     77   7 7 00   0 0  340  21013  1214  1 22   2 2 55   5 5     11   1 1 0  000    1  111 1 111    2 222 3402101312141    3  333    4  444 7  777    0  000

 3402101312141   12 16 0 8 4 0 4 12 4 8 4 16 4   912063039363123    21 25 0 14 7 0 7 21 7 14 7 28 7   0000000000000    6 804 202 6 2 4 2 8 2 15 20 0 10 5 0 5 15 5 10 5 20 5     3402101312141     0000000000000  3402101312141    3402101312141    6 804 202 6 2 4 2 8 2    912063039363123  12 16 0 8 4 0 4 12 4 8 4 16 4     21 28 0 14 7 0 7 21 7 14 7 28 7     0000000000000

112 Neutrosophic Super Matrices and Quasi Super Matrices

Clearly the major product of two super vectors is a super matrix.

Now we show by examples the structure of a major product of a super column vector with its transpose.

Example 3.1.23: Let

 3     0   1     5  1    2   3  A =    7     0   2     1   5     0     1 

be a super column vector. To find the major product of A with At.

Here At = [3 0 1 5 | –1 2 3 7 0 2 1 | 5 0 1] Super Bimatrices and their Generalization 113

 3     0   1     5  1    2   3  Now AAt =   [3 0 1 5 | –1 2 3 7 0 2 1 | 5 0 1]  7     0   2     1   5     0     1 

 33  3      00 0  3015 1237021  501  11  1       55  5   11    1     22 2     33   3     773015 1237021  7 501     00   0     22 2        11   1    55 5      0 3015 01237021 0 501       11 1      114 Neutrosophic Super Matrices and Quasi Super Matrices

9031536 92106 31503  00000000000000 301 5 12 3 702 1 501  15 0 5 25 5 10 15 35 0 10 5 25 0 5 30 1 5 1 2 3 70 2 1 50 1  6021024 61404 21002 9031536 92106 31503   21 0 7 35 7 14 21 49 0 14 7 35 0 0  00000000000000 6021024 61404 21002  301 5 12 3 702 1 501 15 0 5 25 5 10 15 35 0 10 5 25 0 5  00000000000000  301 5 12 3 702 1 501

We see the major product of A with its transpose is a symmetric super matrix.

The symmetry in them occurs in a very special way the main diagonal component is a symmetric matrix where as the other components are transpose of each other; this can be observed from the above super matrix.

Super Bimatrices and their Generalization 115

Example 3.1.24: Let

3   1 2   0 1   4 A = 1   0   2 3   4 5   6

be the super column vector.

To find the major product of A with its transpose.

At = [3 | 1 2 | 0 1 4 | 10 2 3 4 5 6].

We see the major product of A and At results in a special symmetric super matrix whose natural order as well as the matrix order is a square matrix.

116 Neutrosophic Super Matrices and Quasi Super Matrices

3   1 2   0 1   4 We find AAt. AAt = 1 [3 | 1 2 | 0 1 4 | 10 2 3 4 5 6].   0   2 3   4 5   6

33 312  3014  31023456   11   1  1 3  12   014    1023456 22 2 2     00   0  0     13  1 12  1  014   1  1023456 44   4  4  = 11   11    00   00 22   2 2     33312  3014 3 1023456  44   4 4     55   5 5     66   66 Super Bimatrices and their Generalization 117

93603123069121518  312014102 3 4 5 6 624028204 6 81012  000000000 0 0 0 0 312014102 3 4 5 6  12 4 8 0 4 16 4 0 8 12 16 20 24 = 312014102 3 4 5 6  000000000 0 0 0 0  624028204 6 81012 93603123069121518  12 4 8 0 4 16 4 0 8 12 16 20 24 15 5 10 0 5 20 5 0 10 15 20 25 30  186120624601218243036

It is easily observed that AAt is a super symmetric matrix.

This type of minor and major products in case of super vectors are extended to super bivectors which will be exhibited by one or two examples.

Example 3.1.25: Let

A = [0 1 2 1 | 3 4 | 0 2 1]  [1 0 0 0 1 1 | 0 1 2 | 4 1] and

118 Neutrosophic Super Matrices and Quasi Super Matrices

0   11   21     31   40     B = 00    11     12     03   10    1

be two super bivectors. The minor product of A = A1  A2 with B = B1  B2 is given by

AB = (A1  A2) (B1  B2)

= A1 B1  A2 B2

0   1 1   2 1     3 1   4 0     = [0 1 2 1 | 3 4 | 0 2 1] 0  [1 0 0 0 1 1 | 0 1 2 | 4 1] 0   1 1     1 2     0 3   1 0   1 Super Bimatrices and their Generalization 119

1  1 20   = 0121 34 021 0  31       1 4

0  1  1 10   100011 012 2 41  11  3 0  0

= {12 + 4 + 1}  {0 + 8 + 1}

= {17}  {9}.

We give yet another example.

Example 3.1.26: Let

A = [0 1 2 3 4 | 5 6 | 7 | 8 9 0 1 2 3 4]  [ 0 1 2 | 0 1 | 3 0 1 5] and 120 Neutrosophic Super Matrices and Quasi Super Matrices

0  1 0  11    50   10    05   B = 20     12  01    10    02   2  1 0

be two super bivectors.

To find the minor product of A with B.

(In case of minor product we see each of the component submatrix of the row super matrix is multiplied by the corresponding component submatrix of the column super matrix we get the resultant to be a singleton bimatrix, which is clearly not a super matrix).

Super Bimatrices and their Generalization 121

 0     1   0     1   5     1   0    AB = [0 1 2 3 4 | 5 6 | 7 | 8 9 0 1 2 3 4]  2      1   0     1   0     2     1  0 

1   0 0   5 [0 1 2 | 0 1 | 3 0 1 5] 0   2 1   0   2

122 Neutrosophic Super Matrices and Quasi Super Matrices

 1    00   11  =  1   0123400 56 7 2 8901234    0 12     51     0  2 1    51   012 0 01 3015   00    0   2 = {24 + 5 + 14 + 9}  {0 + 0 + 16} = {52}  {16}.

Now we proceed onto illustrate the major product of two super bivectors in an analogous way as we have done for super vector by some examples.

1   0 0    1 1   2 2  3 3   4 1  0 Example 3.1.27: Let A = 4     1 5    5 6    0 7   1 0  0 1   1   0 Super Bimatrices and their Generalization 123

and B = [ 1 0 | 2 4 3 1 | 0 1 2 3 0 1 0]  [3 1 2 0 | 1 1 2 0 0 | 3 0 | 2 5 1 0 1 1] be two super bivectors. To find the major product of these two super bivectors. AB = (A1  A2) (B1  B2) = A1 B1  A2 B2

0   1 2   3 1   = 4 [ 1 0 | 2 4 3 1 | 0 1 2 3 0 1 0]  5   6   7 0   1 1   0 1   2 3   4 0   [3 1 2 0 | 1 1 2 0 0 | 3 0 | 2 5 1 0 1 1] 1   5 0   1 0   1   0 124 Neutrosophic Super Matrices and Quasi Super Matrices

          00   0      101  12431   10123010   22   2    33   3   01   2431   0123010  11   1  44   4      55   5  66   6  01   2431   0123010 77   7  00   0    11   1  

11   11  312 0  112 0 0 30  2 51011  00   00  11   11      22   22  3120 11200 30 251011  33   33      44   44   00   00      13120  111200 130  1251011  55   55   0 000      11 11 312 0  112 0 0 30  2 51011  00   00      11   11 

Super Bimatrices and their Generalization 125

00 0 0 0 000 0 0 000  01 2 4 3 101 2 3 010 02 4 8 6 202 4 6 020  03 6 129 303 6 9 030 01 2 4 3 101 2 3 010  = 04 8 1612404 812040  0 5 10 20 15 5 0 5 10 15 0 5 0  0 6 12 24 18 6 0 6 12 18 0 6 0  0 7 14 28 21 7 0 7 14 21 0 7 0 00 0 0 0 000 0 0 000  01 2 4 3 101 2 3 010

312011200302 51011  000000000000 00000 312011200302 51011  624022400604102022 936033600906153033  12480448001208204044 000000000000 00000  312011200302 51011  15510055100015010255055 000000000000 00000  312011200302 51011 000000000000 00000  312011200302 51011

Thus the major product of two super bivectors is a super bimatrix.

Now we give one more numerical illustration of the major product of two superbivectors 126 Neutrosophic Super Matrices and Quasi Super Matrices

Example 3.1.28: Let

3  1  1 0    2 1    3 1    0 1 1 2    0 0   A =  1 0    0 0    2 6  3 1    2 0 1 0    5 2    1

= A1  A2 and B = B1  B2 = [3 1 0 0 | 1 2 0 5 6 7 | 2 1 0]  [1 0 1 | 2 3 4 0 5 | 1 1 0 0 0 2 1 0 3] be two super bivectors to find the major product of A with B.

In case of the major product of a column superbimatrix with a row row super bimatrix we get the resultant to be a super bimatrix rectangular square.

AB = (A1  A2) (B1  B2) = A1 B1  A2 B2 Super Bimatrices and their Generalization 127

3  1  1 0   2 1 3 1   0 1 1 2   0 0  [3 1 0 0 | 1 2 0 5 6 7 | 2 1 0] 1 [1 0 1 | 2 3 4 0 5 | 1 1 0 0 0 2 1 0 3] 0   0 0   2 6 3 1   2 0 1 0   5 2  1

33  3 13100 1120567  1210 00  0 11  1    113100 120567 1 210 11  1    22  2 00  0 00  0 00  0 66310 0 12 0567  6 210 11  1 00  0 00  0    223100 1205672  210 111 128 Neutrosophic Super Matrices and Quasi Super Matrices

11   1    22   2 33   3    00101  23405  0 110002103 11   1    00   0    11   1 00   0    2101  223405  2110002103 33   3    22   2 101  23405  110002103 11   1  5101 523405  5110002103

9 3003 6 01518216 30  3 1001 2 0 5 6 7 2 10 00000000 0 0 000  3 1001 2 0 5 6 7 2 10 3 1001 2 0 5 6 7 2 10  3 1001 2 0 5 6 7 2 10 6 2002 4 01012144 20  00000000 0 0 000   00000000 0 0 000 00000000 0 0 000  1860061203036421260 3 1001 2 0 5 6 7 2 10  00000000 0 0 000 00000000 0 0 000  6 2002 4 0 10 12 14 4 2 0  3 1001 2 0 5 6 7 2 10 Super Bimatrices and their Generalization 129

101 2 3 4 0 5 11000 2 10 3  202 4 6 8 01022000 4 20 6 303 6 9 1201533000 6 30 9  000 0 0 0 0 0 00000 0 00 0 101 2 3 4 0 5 11000 2 10 3  000 0 0 0 0 0 00000 0 00 0 101 2 3 4 0 5 11000 2 10 3  000 0 0 0 0 0 00000 0 00 0  202 4 6 8 01022000 6 20 6 303 6 9 1201533000 6 30 9  202 4 6 8 01022000 4 20 6 101 2 3 4 0 5 11000 2 10 3  50510152002555000105015

We see AB the minor product of the two super bivectors is a super bimatrix.

Now we proceed on to illustrate the major product of a super column vector with its transpose, by some numerical examples.

We see the major product of a column super vector with its transpose which is a row super vector get the resultant to be a square symmetric super matrix.

130 Neutrosophic Super Matrices and Quasi Super Matrices

Example 3.1.29: Let 2   1 0   1 3   5 0 A =   1   0 3   2 1   7

be a super column vector. To find the major product of A with its transpose At = [2 1 0 1 | 3 5 0 1 0 | 3 2 1 | 7].

2  1 0  1 3  5 t 0 AA =  [2 1 0 1 | 3 5 0 1 0 | 3 2 1 | 7] 1  0 3  2 1  7 Super Bimatrices and their Generalization 131

22  22      11 11 2101 35010 321  7 00  00      11  11   33  33      55 55      002101 35010 00 321  7     11 11     00  00   3333     22101 235010 2321 27     11 11   72 1 0 1 73 5 0 1 0 73 2 1 [7][7]

 420261002064214    21013 50103 217  0 000 0 0 000 0 0 0 0    21013 50103 217  63039150309 6321   1050515250501510535  0 000 0 0 000 0 0 0 0 = . 21013 50103 217  0 000 0 0 000 0 0 0 0 63039150309 6321  420261002064214 21013 50103 210  1470721350702114749

It is easily verified the major product of a super column vector with its transpose is a super matrix. 132 Neutrosophic Super Matrices and Quasi Super Matrices

Having just seen the major product of a super column vector with its transpose now we proceed on to illustrate the major product of a super column bivector with its transpose by a numerical example in an analogous way.

Example 3.1.30: Let

2   1 1   0 2     1 3   3 0     1 1   A = A  A = 0  4 1 2     5 0     7 5   1 1     2 7   0 3     1

be a super column bivector.

t tt A = AA12 = [2 1 0 1 | 3 1 0 | 5 7 1 2 0 1]  [1 2 | 3 0 1 4 | 0 5 1 | 7 3].

t tt The major biproduct of AA = (A1  A2) ( AA12 ) which will result in a symmetric square super bimatrix. Super Bimatrices and their Generalization 133

2   1 0   1 3   1 = AAtt AA = 0 [2 1 0 1 | 3 1 0 | 5 7 1 2 0 1]  11 2 2   5   7 1   2 0   1

1   2 3   0 1   4 [1 2 | 3 0 1 4 | 0 5 1| 7 3] 0   5   1 7   3

134 Neutrosophic Super Matrices and Quasi Super Matrices

    222     111 2101 310  571201  000     111   333      12101 1310  1571201  000   555     777  111  2101 310  571201 222  000        111 

11   1  1 12   3014   051    73 22   2  2 33   3  3     00   0  0 12  3014   051   73 11   1  1     44   4  4  00   0  0     512  53014   5051    573  11   1  1  77  77  12 3014 051  73 33  33  Super Bimatrices and their Generalization 135

 420262010142402    21013105 71201  00000000 00000    21013105 71201  630393015213603    21013105 71201   00000000 00000   105051550253551005   147072170354971407 21013105 71201  420262010142402 00000000 00000  21013105 71201

12 30140517 3  24 60280102146 36 903120153219  00 00000000 0 12 30140517 3  4 812041602042812. 00 00000000 0  51015052002553515  12 30140517 3 71421072803574921  36 903120153219

We see the major product of a super column bivector with its transpose a super bimatrix which is symmetric.

136 Neutrosophic Super Matrices and Quasi Super Matrices

Example 3.1.31: Let 17   03 16   00 21   12 A = 31   00   10 33   73 51   11 = A  A be a super column bivector. At = AAtt = [ 1 0 1 0 1 2 12 | 2 1 3 | 0 1 3 7 5 1]  [7 3 6 0 –1 2 1 | 0 0 3 | 3 1 1]; we find t tt tt AA = (A1  A2) ( AA12 ) = AA11 AA 2 2 1  0 1  0 2  1 = 3 [ 1 0 1 0 | 2 1 3 | 0 1 3 7 5 1]   0  1 3  7 5  1 Super Bimatrices and their Generalization 137

7  3 6  0 1  2 1 [7 3 6 0 –1 2 1 | 0 0 3 | 3 1 1]  0  0 3  3 1  1

    111     000 1010 213  013751  111     000   222      11010 1213  1013751  333   000     111  333  1010 213  013751 777  555     111     138 Neutrosophic Super Matrices and Quasi Super Matrices

77 7    33 3 66 6    007360 121 003 0 311 11 1    22 2    11 1 00 0    07360 121 0003  0311 33 3 33 3    17360 121 1003 1311 11  1

1010 2 1 3 01 3 7 5 1   0000 0 0 0 00 0 0 0 0 1010 2 1 3 01 3 7 5 1   0000 0 0 0 00 0 0 0 0 202042602614102   1010 2 1 3 01 3 7 5 1 = 3030 6 3 9 03 9 21153   0000 0 0 0 00 0 0 0 0   1010 2 1 3 01 3 7 5 1 3030 6 3 9 03 9 21153  707014721072149357 505010515051535255  1010 2 1 3 01 3 7 5 1 Super Bimatrices and their Generalization 139

49 21 42 0 7 14 7 0 0 21 21 7 7  21 9 18 0 3 6 3 0 0 9 9 3 3 42 18 24 0 6 12 6 0 0 18 18 6 6  0000000000000 73601  21003311   14 6 12 0 2 4 2 0 0 6 6 2 2 7360121003311  0000000000000  0000000000000 21 9 18 0 3 6 3 0 0 9 9 3 3  21 9 18 0 3 6 3 0 0 9 9 3 3 7360121003311  73601 21003311

It is easily seen AAt is a symmetric super bimatrix.

3.2 Operations on Super bivectors

In this section we define the notion of super row bivector of type B and super column bivector of type B. We define the basic operations of addition, subtraction, minor and major product whenever such operations are compatible. This concept is made easy to understand by numerous examples.

Now we proceed onto define the new notion of super row bivector of type B and super column bivector of type B.

DEFINITION 3.2.1: Let A = A1  A2 where A1 and A2 are distinct super row vectors of type B then we call A to be a super row bivector of type B (super birow vector of type B).

We first illustrate this by an example. 140 Neutrosophic Super Matrices and Quasi Super Matrices

Example 3.2.1: Let

3013112345  A = 1100101020 5202030405

1012331211  0101100223   = 0512011401  1000102010

A1  A2. A is a super row bivector of type B.

Example 3.2.2: Let

120010311234 A =   315501020123

1001000  2120111 3000001

= A1  A2; we see A is a super row bivector of type B.

Example 3.2.3: Let A = A1  A2 =

00000000  00000000   00000000  00000000

000000000  000000000; 000000000 Super Bimatrices and their Generalization 141

A is a super row bivector of type B. We see A is a zero super row bivector of type B.

Example 3.2.4: Let A = A1  A2 =

11111111111  11111111111 11111111111   11111111111 11111111111

111111111   111111111 111111111

be a super row bivector of type B.

We call this as a super unit row bivector of type B or super row unit bivector or bisuper unit vector of type B.

We now proceed onto define the notion of super column bivector of type B.

DEFINITION 3.2.2: Let A = A1  A2 where A1 and A2 are distinct super column vectors of type B. We call A to be a super column bivector of type B.

We illustrate this by the following examples.

142 Neutrosophic Super Matrices and Quasi Super Matrices

Example 3.2.5: Let 301   30145 458  21037 767   12345 841  54321 110    A = A1  A2 = 01010  202 ;  10101 010   12301 123   40512 456  37200 789   012

A is a super column bivector of type B.

We give yet another example before we proceed to give examples of the notion of super zero column bivector and super column unit bivector of type B.

Example 3.2.6: Let

31   1230 21  1110 12   0001 13  A = A  A = 2345  30 ; 1 2   6789 10  3102 11    2011 01   51

A is a super column bivector of type B. Super Bimatrices and their Generalization 143

Example 3.2.7: Let

00   00 000 00   000 00 000 00   000 00 A = A  A = 000  00 ; 1 2   000 00   000 00   000 00   000 00 00   00

A is defined to be a super zero column bivector of type B.

Example 3.2.8: Let

000 000     000 000  000 000     000 000  000 000  A = A1  A2 =    ; 000 000  000 000     000 000     000 000  000 000 

A is not a super zero column bivector of type B as A1 = A2. 144 Neutrosophic Super Matrices and Quasi Super Matrices

Example 3.2.9: Let A = A1  A2 =

0000 000000 0000   000000 0000   000000 0000   000000 0000  000000 ; 0000   000000 0000 000000 0000   000000 0000   000000 0000

A is a super zero column bivector of type B.

Example 3.2.10: 1111   111 1111  111 1111   111 1111  111 1111 Let A =    ; 111 1111  111 1111   111 1111   111 1111 1111

A is a super unit column bivector of type B.

Now as in case of super row bivectors of type B we can define the transpose of a super row bivector of type B. It is easy to see that the transpose of a super row bivector of type B is a super column bivector of type B and vice versa. Super Bimatrices and their Generalization 145

We shall illustrate this situation by some simple examples.

Example 3.2.11: Let A = A1  A2 =

3120301013  1011410100  0120512345

3110345   1111234   0100567   1126011 be a super row bivector of type B.

Now transpose of A denotes

310  101 3101 212   1111 010     1102 t t tt345   by A = (A1  A2) = AA12 0106 011 . 3250 102   4361 013   5471 104 305

It is easily seen that At is a super column bivector of type B.

146 Neutrosophic Super Matrices and Quasi Super Matrices

Example 3.2.12: Let 21   35 013   67 111   01 021 12 456   34 701 56    A = 301 01 084   23 011   45 110   67 101 89 111   12   03

be a super column bivector of type B.

The transpose of A denoted by At =

01047300111 tt A A 11250081101 12 31161141011

23601350246810 . 15712461357923

Clearly At is a super row bivector of type B.

Now as in case of type A super row (column) bivectors we can define identical in structure super row (column) bivectors of type B.

Super Bimatrices and their Generalization 147

DEFINITION 3.2.3: Let A = A1  A2 where A1 and A2 are distinct i.e., A = A1  A2 is a super row bivector of type B. Let B = B1  B2 be a super row bivector of type B.

We say A and B similar or “identical in structure” if the length of Ai and Bi are the same and the number of rows in both Ai and Bi are equal and the partitions in Ai and Bi are identical for i=1,2.

By the term partitions are identical we mean if in Ai the first th partition is say between the r and (r+1) column then in Bi also the first partition will be only between the r and (r+1)th column and the number of partition carried out in Ai will be the same as that carried out in Bi, i=1,2.

We will illustrate this by some simple examples.

Example 3.1.13: Let

301237120132   A = A1  A2 = 110101234015 000511000178

23231254   31002310    and 15000000   01711237

110012003421  B = 010000560001  000170000810

148 Neutrosophic Super Matrices and Quasi Super Matrices

10111001  21001110  31110110  50001111

be two super row vectors of type B. We see A and B are “identical in structure”. For the bilength of A and B is (12, 8).

Also the partitions are identical in Bi and Ai, i=1,2. The number of rows in Ai equal to the number of rows in Bi, i=1,2.

Example 3.2.14: Let

200113378 A = A1  A2 =    116215123

57838800513  91241168157 and 03730074278

135678438 B = B1  B2 =    249012259

10279893155  34501203577 67862007119

be two super row bivectors of type B. We see A and B “identical in structure”. Super Bimatrices and their Generalization 149

Example 3.2.15: Let

31237817 11   57910158 2 2 A = (A1  A2) =  01271569 3 3   11012340 4 4

3417017084  0700011256 1008623708

1111013213   and B = B1  B2 = 2020826127  3021100136

213710438  487623675 be two super row bivectors of type B. We see A and B does not have identical structure.

We define now addition of two super row bivectors of type B of identical structure.

DEFINITION 3.2.4: Let A = A1  A2

111111 aaaaaa 11 1r 1r 1 1s 1t 1m =   111111 aaaaaan1 nr nr 1 ns nt nm

bbbbb11111 11 1u 1r 1s 1w  11111 bbbbbp1 pu pr ps pw 150 Neutrosophic Super Matrices and Quasi Super Matrices

and if B = B1  B2 =

222222 aaaaaa 11 1r 1r 1 1s 1t 1m   222222 aaaaaan1 nr nr 1 ns nt nm

bbbbb22222 11 1u 1r 1s 1w  22222 bbbbbp1 pu pr ps pw

are two super row bivectors of type B. We see A and B are identical in structure. The sum of A and B denoted by A+B = (A1  A2) + (B1  B2) = A1 + B1  A2 + B2

12 121 2 12 12 12 aa aaa a aa aa aa 11 11 1r 1r 1r 1 1r 1 1s 1s 1t 1t 1t 1t  12 121 2 12 12 12 aan1 n1 aaa nr nr nr 1 a nr 1 aa ns ns aa nt nt aa nt nt  12 12 1 2 12 1 2 bb bb b  b  bb  bb 11 11 1u 1u 1r 1 1r 1 1s 1s 1w 1w  12 12 1  2  12  12 bbp1 p1 bb pu pu bpr 1 b pr 1 bb ps ns bbpt pt

is defined to be the sum of the two super row bivectors A and B of type B. Clearly A+B is again a super row bivector of type B with identical structure with A and B.

Now in view of this we have the following interesting theorem.

THEOREM 3.2.1: Let X = {A = A1  A2 be the collection of all super row bivector of type B which are identical in structure with A with entries from the field Q or R or C or Zp (p a prime) or from the rings Z or Zn, n not a prime or C(Zp) or Z(g1) or Super Bimatrices and their Generalization 151

2 Z(g2) or Z(g3) (g1, g2 and g3 are new elements such that g1 = 0, 2 2 g2 = g2 and g3 = –g3 and Z can be replaced by Q or R or C or

C(Zn) or Zn}. Then X is a group under addition.

Example 3.2.16: Let X = X1  X2 =

3210 10013 4 0 1  1526 0 1231 0 0 1  0630 6 1000 10 0

31414130021011  and 632 120111020 11

3 1 0010110 40 1  Y = 0 12010201 0 0 1  0 1 3101010 1 1 1

30 1 0301012 1011  21 110201011101

= Y1  Y2 be two identical structure with the super row bivector of type B. To find the sum

X+Y = (X1  X2) + (Y1  Y2) = (X1 + Y1)  (X2 + Y2)

031000123000  = 144611432000  076162010011

0 1517 1401400 2 2 . 8 4 102 2 121121 12

We see X+Y is again a super row bivector with identical structure as that of X and Y. The subtraction of X with Y 152 Neutrosophic Super Matrices and Quasi Super Matrices

X–Y = (X1 – Y1)  (X2 – Y2)

611020103802  = 160 6 11 0 3 0 0 0 2 050 1 6 0 0 10 2 1 1

6 1311 120 10 20 00 . 423 22 2101  12  1 10

X–Y is again a super row bivector of type B.

Now in a similar way we can define the notion when are two super column bivectors of type B are “identical in structure”. Let A = A1  A2 and B = B1  B2 be two super column bivectors of type B. We say A and B are super column bivectors of type B are identical in structure if the number of natural rows in Ai is equal to Bi, i=1,2 and the number of columns in Ai equal to Bi and the partition of Ai and Bi are identical. We can as in case of identical structure super row bivectors of type B, we can in case of identical structure of super column bivectors of type B, we can define addition and subtraction of super column bivectors of type B.

Example 3.2.17: 310 0 328 001 1 060  402 1 110 320 0 101  10 12 641 123 4 001 Let A =  and 670 2 702 023 1 080 10 1 1 103  425 6 462 012 0 675  018 3 320 Super Bimatrices and their Generalization 153

 010 0 012      001 0 020   10 10 110      010 1 010   110 2 011      003 6 110  B =   112 4 102      001 3 063     10 0 0 1 0 1   201 1 215      800 1 0 10   011 4 025  be two super column bivector of type B. The sum of A+B = (A1 + B1)  (A2 + B2)

32 0 0 3 310  00 2 1 0 8 0 50 1 1 2 2 0  33 0 1 1 1 1 11 1 4 6 5 2  12 6 10 1 1 1  78 2 6 8 0 4  02 4 4 0143  20 1 1 2 0 4 62 6 7 6 7 7  81 2 1 6 6 5 02 9 7 3 4 5 is again a super bicolumn bivector of type B.

In view of this we have the following interesting theorem. 154 Neutrosophic Super Matrices and Quasi Super Matrices

THEOREM 3.2.2: Let V = {X = X1  X2 / collection of all super column bivector of type B with entries from a field or ring which are identical in structure}. Then V is a group under addition.

However to define multiplication we have to define either minor product or major product.

We proceed onto give the minor product of two super bivectors of type B by numerical illustrations.

Example 3.2.18: Let

15325701   X = (X1  X2) = 06211011  27100101

12010112340  01102101001  01010500112  10103011001

1201   0115 310   0010 100   2100 001 1001 002   and Y =  0110 = Y1  Y2 101 1001 221   0101 453   1010 012 0201   0120 Super Bimatrices and their Generalization 155

be two super bivectors.

The minor product of X with Y is given by XY = (X1  X2) (Y1  Y2) = X1 Y1  X2 Y2.

310   100 001 15325701  002 = 06211011   101 27100101  221 453   012

1201   0115 0010   2100 12010112340 1001 01102101001  0110 01010500112 1001 10103011001  0101   1010 0201   0120

153310 2  = 062100 1002 271001 0

156 Neutrosophic Super Matrices and Quasi Super Matrices

101 12011201 5701 221 01100115 1011 453 01010010 0101 012 10102106

1001 01 12340     0101 211001  01001   1010 050110  00112     0201 30 11001  0120

8 1 3 0 0 4 19 15 14  = 602 002 5 6 6 13 2 1 0 0 0 2 3 3

0110 3521 41037  2112 0125 0 2 21  0550 2215 1 4 51  3003 1211 1 2 22

7166 8 27 16 21 24 58 = 11 6 10    . 311116 15 5 4 54 35

It is pertinent to mention that the entries of these super matrices can be replaced by neutrosophic rings, special dual like number neutrosophic rings, special quasi dual number neutrosophic rings and mixed special dual number neutrosophic rings.

Chapter Four

QUASI SUPER MATRICES

A square or a rectangular matrix is called as a quasi super matrix. It cannot be obtained by partition of a matrix. All super matrices are quasi super matrix but quasi matrix in general is not a super matrix. Thus the elements of a quasi matrix are submatrices following some order.

DEFINITION 4.1: Take a m × n matrix A with entries from real or complex numbers.

 AA11 12  A1r   1   AA21 22  A2r  A = 2      AA A  s1tt s2 sr tm

where Aij are submatrices of A with 1  j  r1 , r2, …, rm and 1  i  s1, s2, …, st.

158 Neutrosophic Super Matrices and Quasi Super Matrices

Clearly we do not demand the size of Aji to be same but the only condition to be satisfied is that if A is a m × n matrix then the number of the rows in each column of the matrix adds up to m and the number of the columns in each of rows added up to n. We define A to be a quasi super matrix.

Further we do not demand r1 = r2 and ri = rj in general also. Also si  sj in general.

We give examples of one or two quasi super matrix before the proceed on to define some of their basic properties.

Example 4.1: Let A be a 6 × 4 quasi super matrix given by

321 2  015 0

983 2 AA11 12  A = =   141 4 AA21 22  253 0  315 1

where

321 2 A11 =   , A12 =   , 015 0

98 32     14 14 A21 = and A22 = . 25 30     31 51 

We see the sum of the rows of A11 and A21 adds up to 4 and sum of the rows of A12 and A22 adds up to 6 where as some of columns adds up to 4 i.e., the sum of the columns of A11 and A12 is 4 and that of A21 and A22 is also 4. Quasi Super Matrices 159

Next we give some more examples.

Example 4.2: Let A be 9 × 8 quasi super matrix given by

  2031 12 30  0521 21 11  3120  1 310  1211 0 A = 031  3300  2 202  5111 1  303 012  2 0    110  110  1 4 101

AAA11 12 13    = AAA21 22 23    AAAA31 32 34 33 

Clearly the number of columns added up in A11 A12 and A13 to 4 + 2 + 2 = 8.

The sum of the columns added up in A21 A22 and A23 to 4 + 3 + 1 = 8. The sum of the columns of A31 A32 A34 and A33 to 3 + 1 + 3 + 1 = 8.

We see basically A is a 8 × 8 matrix but by the division it is made into a quasi super matrix of the given form having 10 number of submatrices. A11, A12, A13, A21, A22, A23, A31, A32, A33 and A34.

Now any 8 × 8 matrix can form a division in the following form given in the example 4.3.

160 Neutrosophic Super Matrices and Quasi Super Matrices

Example 4.3: Let A be any 8 × 8 matrix.

52102   031 11021  152 00110   001 52001  11220 A =  19312  01010 120  10101 315  22025 115  31020

AA11 12  =   . AA21 22 

Here the sum of the rows of the sub matrices A11 and A21 is 3 + 5 = 8 and sum of the columns of the sub matrices A21 and A22 is 5 + 3 = 8. We make the following observation we see all the four sub matrices are square matrices based on these observations we make the following definition.

DEFINITION 4.2: Let A be the m × m square matrix. If all the sub matrices of A are also square matrices then we call A to a quasi super square matrix.

The matrix given in example 4.3 is a square quasi super matrix. We give yet another example before we define some more results.

Example 4.4: Let A be a 8×8 matrix with the following representation.

Quasi Super Matrices 161

031521 02  01 152110  001001 10  793121 01 A =  010101 20  101013 20  22 02  53  15       31 02  01  15 

 A12    AA =  11 22   A   23  AAAA21 31 24 25 

Sum of the rows A11 and A21 = 8, Sum of the rows A11 and A23 = 8, Sum of the rows A11 and A24 = 8, Sum of the rows A12, A22, A23, A25 is 8, Sum of the columns of A11 and A12 is 8, Sum of the columns of A11 and A22 is 8, Sum of the columns of A11 and A23 is 8, Sum of the columns of A21, A31, A24 and A25 is 8.

Example 4.5: Let A be a 5 × 6 matrix which is given in the following.

301 561   110 325   A = 212 102    21 01  11      02 10  11 

162 Neutrosophic Super Matrices and Quasi Super Matrices

AA11 12  =   AAA21 22 23 

where A11, A12, A21, A22 and A23 are submatrices.

Clearly sum of rows of A11 and A21 is 5, sum of rows of A12 and A22 is five and that of A12 and A23 is five.

Sum of the columns A11 and A12 is six. Sum of the columns of A21 A22 and A23 is six.

Example 4.6: Let A be a 6 × 6 quasi super matrix with

AA11 12   00 A =  AA ,   AA21 22 

where A11 is a 4 × 4 square matrix and A12 is a 2 × 2 square matrix. A21 is a 2 × 2 square matrix and A22 is a 4 × 4 square matrix.

Now

031 2   110 5 A11 = , 512 1   010 1

01 A12 =   , 21

11 00 A21 = , A° = A° =   and 02 00

Quasi Super Matrices 163

6012   1002 A22 = . 1100   0101

031201   110521 512 100   010100 A =   . 006012   001002 111100    020101

We see all the six sub matrices are square matrices.

Example 4.7: A be a 4 × 5 matrix which is a quasi super matrix given by

 010 20   100 10 A =  111   11  101 01 

AA11 12  =   . AA21 22 

Similarly we proceed on to define the notion of quasi super matrix which is rectangular.

DEFINITION 4.3: Let A be a quasi super matrix.

164 Neutrosophic Super Matrices and Quasi Super Matrices

 AA A 11 12 1r1    AA21 22  A1r  A = 2      AA A  s112 s 2 sr tm

Aij’s are rectangular matrices, with 1 < i < r1, r2, …, rm and 1 < j < s1, s2,…, st, we define A to be a quasi super rectangular matrix.

We give an example of a matrix of this type.

Example 4.8: Let A be a 6 × 8 matrix quasi super matrix. 2012 1100  1105 1011 110 50111  A =   101 11010   111 10001    201 22022 

AA11 12  =   . AA21 22 

A is a rectangular quasi super matrix.

Example 4.9: Let A be a 6 × 4 matrix. 210 1   112 0 120 1   A =     1025     3102     1110  Quasi Super Matrices 165

AA11 12  =    A21  which is a quasi super matrix. Sum of the columns of A11 and A12 is four.

Number of columns in A21 is four. Sum of rows of A11 and A21 is six. Sum of rows of A12 and A21 is six. The quasi super matrix described in example 4.9 is not a rectangular quasi super matrix for it contains a 3 × 3 square matrix in it.

We proceed on to define these type of matrices,.

DEFINITION 4.4: Let A be a m × n rectangular matrix. A be a quasi super matrix.

 AA11 12  A1r   1   AA21 22  A1r  A = 2      AA A  s112 s 2 sr tm

If some of the Aij’s are square submatrices and some of them are rectangular submatrices, then we call A to be mixed quasi super matrix.

Thus we have seen 3 types of quasi super matrices.

We have to give some more examples for we can have a square m × m matrix A: yet A can be a mixed quasi super matrix or a rectangular quasi matrix. Likewise a square m × m matrix can be a mixed quasi super matrix.

Example 4.10: Consider a 6 × 6 quasi super matrix where

AA11 12  A =   AA21 22  166 Neutrosophic Super Matrices and Quasi Super Matrices

where

 4102   12  A11 =     = A12  6025   21 

312 101    101 010 A21 = and A22 = 010 111    210 022

201421   520612 312101 i.e., A =   . 101010 010111   210022

A is a 6 × 6 matrix but A is not a rectangular quasi super matrix or a square quasi super matrix it is only a mixed quasi super matrix.

Thus this example clearly shows a quasi super matrix which is a 6 × 6 square matrix can be a quasi super mixed square matrix.

Next we proceed on to give an example of a square matrix which is only a rectangular quasi super matrix.

Example 4.11: Let A be a 5 × 5 quasi super square matrix.

Quasi Super Matrices 167

01234   12340 A = 23401   34012   40123

AA11 12    = AA21 22     A51  where

34 012   A11 =   , A12 = 40 123 01

234 A21 =   , A22 = [ 1 2 ] and 340

A31 = [ 4 0 1 2 3 ].

A is a quasi super rectangular matrix.

Example 4.12: Let A be a 6 × 4 quasi super rectangular matrix.

0123   1102 

2010 AA11 12  A =   =   . 1671  A21  5208   3192

168 Neutrosophic Super Matrices and Quasi Super Matrices

Clearly A is a square quasi super matrix which is not a square matrix. We see all the three submatrices of A are square matrices.

Now having seen three types of quasi super matrices, we now proceed on to define some sort of operation on then. First given any n × m matrix which we are not in a position to give a partition as in case of super matrices.

So we define in case of n × m matrices a new type of relation called quasi division.

A quasi division on the rectangular array of numbers is a cell formation such that the cells are either square, rectangular or row or column or singletons. Clearly or is not used in the mutually exclusive sense.

We just illustrate this by a very simple example.

Example 4.13: Let A be a 7 × 6 quasi super matrix.

 078941    101010 11 1101   A =  801210 .  011101    21 1400    100111

We see the matrix or this 7 × 6 array of numbers which have been divided in 10 cells. Each cell is of a varying size. For we see first cell is a 2 × 2 square matrix.

07   = I cell = A11 10 Quasi Super Matrices 169

The second cell is a rectangular 2 × 3 matrix.

894   is the II cell-A12 101

1  is the III cell - A13 is a 2 × 1 column vector or matrix. 0

1   The fourth cell is again a column vector given by  8  = A21  0 

The fifth cell is [1], a singleton A22. The sixth cell is [–1 1 0 1] is a row vector A23.

2 The seventh cell is a column vector   = A31. The eighth cell is 1 a 4 × 3 rectangular matrix.

012   111 = A32 114    001

The ninth cell is a 3 × 2 rectangular matrix.

10   01 = A33. 00

th The 10 cell find its place as A44 a row vector [11]. The arrangement of writing or notational order is written in this form 170 Neutrosophic Super Matrices and Quasi Super Matrices

AAA11 12 13    AAA A =  21 22 23  AAA  31 32 33   A44 

We see the sum of the columns of A11, A12 and A13 is 6. Sum of the columns of A21, A22 and A23 is six.

Sum of the columns of A31, A32 and A33 is six. (1+3+2=6). Also sum of the columns of A31, A32 and A44 is six.

Now having seen as example we define the notion of cell partition.

DEFINITION 4.5: A cell partition of m × n rectangular array of numbers A is a division of the m × n array of numbers using cells. The cells can only be singletons or row vectors or column vectors or square matrix and (or) rectangular matrix. i.e., each cell can be called as a sub matrix of A.

For instance this process is like finding subsets of a set. In that case we have lots of choices and we know given N number of elements in the set; the number of subsets is 2N. But dividing the n × m array of numbers of the n × m matrix is not identical for the cell division is not arbitrary each cell should have a form and the division can be many for instance we will first illustrate in how many ways a 2 × 2 matrix can be divided using the method of cell division,

01 Example 4.14: Let A =   23

We enumerate the number of cell division leading the quasi super matrix.

01 A11  A1=   =  A12  , 23 A21  Quasi Super Matrices 171

01 AA11 12  A2 =   =   , 23  A22 

01 A3 =   23

01 and A4 =   23

There are only 4 quasi super matrices constructed using a 2 × 2 matrix.

However we have several partitions leading to a super matrix.

01 B =  is a partition, 23

B is a super matrix.

01 C =  is a super matrix different from B. 23

01 D =  is a super matrix different from both B and C. 23

Thus using 2  2 matrices we have only 3 super matrices. But using cell division leading to quasi super matrix we have 4 quasi super matrices. Thus we have altogether 7 quasi super matrices if we make the rule all super matrices are quasi super matrices. That is the class of super matrices is contained in the class of quasi super matrices.

172 Neutrosophic Super Matrices and Quasi Super Matrices

Example 4.15: Let

012   A = 101 111 be a 3 × 3 matrix.

The number of quasi supermatrices from A are

012   AA11 12  A1 = 101 =    A22  111

012   AA11 12  A2 = 101 =    A22  111

012  A12      A3 = 101 = AA11 22      111  A13 

012  A13  A4 = 101 = AA11 12   A23  111

012  A13  A5 = 101 = AA11 12   A23  111

012 AAA11 12 13    A6 = 101 =  A23    111  A33  Quasi Super Matrices 173

012   AAA11 12 13  A7 = 101 =    A22  111

012   AAA11 12 13  A8 = 101 =    A22  111

012 AAA11 12 13      A9 = 101 =  A22      111  A32 

012   AAA11 12 13  A10 = 101 =    AA22 23  111

012   AAA11 12 13  A11 = 101 =    AA22 23  111

012 AAA11 12 13      A12 = 101 =  AA22 23      111  A33 

012   AAA11 12 13  A13 = 101 =    AA22 23  111

174 Neutrosophic Super Matrices and Quasi Super Matrices

012 AAA11 12 13  A14 = 101 =    AA22 23  111

012 AAA11 12 13    A15 = 101 =  AA22 23    111  A33 

012 AAA11 12 13    A16 = 101 =  AA22 23    111  A32 

012 AAA11 12 13    A17 = 101 =  AA22 23    111  A32 

012 AAA11 12 13    A18 = 101 =  AA22 23    111  A33 

012 AAA11 12 13    A19 = 101 = AAA21 22 23    111  AA32 33  and so on.

Thus we see even in case of a 3 × 3 matrix the number of quasi super matrix is very large. If we make the inclusion of super matrix into it. It will still be larger.

Thus at this juncture we propose the following problem.

Problem: Suppose P is any m × m square matrix. i.e., a square array of m × m numbers. Quasi Super Matrices 175

1. Find the number of super matrix constructed using P. 2. Find the number of quasi super matrix constructed using P.

Now we just try to work with a 2 × 4 matrix.

 3210  Example 4.16: Let M =   be a 2 × 4 matrix.  0101 

We just indicate some of the quasi super matrix constructed using M.

0123  A12  M1 =   = A11  1010  A22 

0123  A13  M2 =   = AA11 12  1010  A23 

0123 AAAA11 12 13 14  M3 =  =   1010  A23 

0123 AAA11 12 13  M4 =  =   1010  AA23 24 

0123 AAA11 12 13  M4 =   =   1010  AA23 24 

0123 AAAA11 12 13 14  M5 =  =   1010  A23 

0123 AA11 12  M6 =   =   1010  A22  176 Neutrosophic Super Matrices and Quasi Super Matrices

0123 AAA11 12 13  M7 =  =   1010  A22 

0123 AA11 12  M8 =  =   1010  AA22 23 

0123 AAA11 12 13  M9 =  =   1010 A21 

and so on. Thus we see even in case of the simple 2 × 4 matrix, we can have several quasi super matrices. We have shown only nine of them.

Now we propose the following problem.

Problem

1. Find the number of quasi super matrices that can be constructed using m × n matrix.

2. Find the number of super matrices that can be constructed using just a m × n rectangular matrix.

Now having defined the notion of cells partition of a matrix which has lead to the definition of quasi super matrices we proceed on the study the further properties of quasi super matrices like matrix addition and multiplication and see how best those concepts can be defined on them.

Example 4.17: Let A be a 9 × 4 rectangular matrix.

Quasi Super Matrices 177

0125   1101 0111   1010 A = 0101   1100 1001   2135   1234

AA11 12    AA =  21 22  .  A   32  AAA41 42 43 

We see A has 8 cells. We can have for the same A we can just have only 4 cells.

0125   1101 0111   1010 AA11 12  A = 0101 =   .   AA21 22  1100 1001   2135   1234

Suppose we have two quasi super matrices how to add? We first proceed on to define addition of a super quasi matrix A with itself. 178 Neutrosophic Super Matrices and Quasi Super Matrices

Example 4.18: Let A be a 5 × 7 quasi matrix with a well defined cell partition defined on it.

0123456  1101101 A = 0110110  5782143  8170 2 5 8

AAA11 12 13    =  A22  .   AAA31 32 33 

 0 2 4681012    2202202 A + A = 2A =  0220220    10 14 16 4 2 8 6 16 2 14 0 4 10 16

2A11 2A 12 2A 13  = 2A22 = 2A.  2A31 2A 32 2A 33

It is left as an exercise for the reader to verify that for the quasi super matrix A we have A – A = [0]. In this case

0000000   0000000 A – A = 0000000 .   0000000 0000000 Quasi Super Matrices 179

Just a quasi super matrix A under the same cell division can be added any number of times and this will not change the nature of a quasi super matrix cell division.

Thus we can say if A is a quasi super matrix then A + … + A: n times is the same as nA.

If A has A11, A12, …, Art to be the collection of all its submatrices then nA will have nA11, nA12, …, nArt to be the collection of all its submatrices.

Note: We can always take any m × n zero matrix and make the cell partition or division in a desired form so that A + (0) = (0) + A = A. For this we first define a simple notion called cell division function.

Suppose we have two m × n matrices A and B. We know the cell division of A how to get the same cell division of B so that the cell divisions of both A and B are the same and both of them are quasi super matrices of same type or similar.

DEFINITION 4.6: Let A be a m × n quasi super matrix. B any m × n matrix. To make B also a quasi super matrix similar to A; define a cell function Fc from A to B as follows.

Fc(A11) = B11 formed from B by taking the same number of rows and columns from B. So that A11 and B11 have the same number of rows and columns, not only that the placing of B11 in B is identical with that of A11 in A.

The same procedure is carried out for every Aij in A.

Thus the cell function Fc converts a usual m × n matrix B into a desired quasi matrix A.

Before we proceed on to define the properties of Fc we show this by some illustrations.

180 Neutrosophic Super Matrices and Quasi Super Matrices

Example 4.19: Let A be a given 3 × 5 quasi super matrix. B be any matrix. To find a similar cell division on B identical with A. Given 03520   A = 14412 25304

AA11 12  =   AAAA21 22 23 24 

is a quasi super matrix with A11, A12, A21, A22, A23 and A24 as submatrices.

Given

0 0.3 4 0.1 2   B = 14020 35111

Define the cell function Fc from A to B as follows.

00.3 Fc(A11) =   = B11, 14

Fc(A12) = [4 0.1 2] = B12,

Fc(A21) = [3 5 1] = B21,

Fc(A22) = [0] = B22,

2 Fc(A23) =   = B23 1

Quasi Super Matrices 181

0 and Fc (A24) =   = B24. 1

0 0.3 4 0.1 2   B = 14020 35111

BB11 12  =   . BBBB21 22 23 24 

We enumerate some of the properties of Fc. Let Fc be a cell division function from A to B where A is a quasi super matrix.

Fc(A) = A

i.e., A is equivalent to A under Fc; i.e., Fc is reflexive.

If Fc(A) = B, i.e.,

A is equivalent to B under Fc then B is equivalent to A under Fc.

i.e., if Fc(A) = B then Fc (B) = A.

(Fc (Fc(A)) = A for all A).

If A is any quasi super matrix and B and C any two m × n matrices. If Fc is the cell division function such that Fc(A)  B so that B becomes a quasi super matrix. Suppose Fc(B)  C so that C becomes a quasi super matrix, then we see A, B and C and quasi super matrices with

A~B and B~C implies A~C.

Thus Fc is an equivalence relation on the class of all m × n matrices under a special or a specified cell division function.

182 Neutrosophic Super Matrices and Quasi Super Matrices

If we take the collection of all cells partitions of a m × n matrix and denote it by C then C is divided into disjoint Fc Fc classes by this relation i.e., C= {F | F ; A  B; A and B Fc c c m × n matrices, A is a quasi super matrix and B any m × n matrix}

We illustrate this by an example.

Example 4.20: Let

ab T =  a, b, c, d  Z2 = {0, 1}}. cd

To find the partition of T under the cell division function.

First we find the number of elements in C. Fc

0 ab Fc =   cd

  1 ab Fc =   cd

2 ab Fc =   cd

  3 ab Fc =   cd

  4 ab Fc =   cd

Quasi Super Matrices 183

  5 ab Fc =   cd

6 ab Fc =   cd

7 ab Fc =   cd

We see C = {F,0 F,1 F,2 F,3 F,4 F,5 F6 , and F,7 Fc c c c c c c c c i.e., | C| = 8. Fc Now we have to find the number of matrices in T.

T =

00 11 10 01 00 00 11 10 ,,,,,,,, 00 11 00 00 10 01 00 10

01 00 10 01 11 11 01 10 , , , ,,,, . 01 11 01 10 10 01 11 11

Thus each and every class in C will have all the 16 Fc elements of T with no over lap.

If C is the class then every class contains exactly the same Fc number of elements in the set of matrices on which the quasi super matrices notion is to be defined.

Now as partition yields to a super matrix we see cell division leads to the concept of quasi super matrix.

Now two m × n quasi super matrix can be added if and only if they have the same cell division defined an them, otherwise 184 Neutrosophic Super Matrices and Quasi Super Matrices

we say addition is not compatible for the cell division is not compatible.

Example 4.21: Let

301 2 5   710 1 1 600 1 0 A =   301 0 0 125 11   116 2 1

AA11 12    = AAA21 22 13  ;   A31 

where A11, A12, A21, A22, A13 and A31 are submatrices of A.

11002  50110 BB11 12  62102   B =  = BB. 11011  21 22    B31  20101  30220

Clearly the addition of A and B is not compatiable for we see the submatrices B11, B12, B21, B22, and B31 are different from A11, A12, A21, A22, A13 and A31.

Thus we see addition of A with B is impossible though both A and B are of same order viz., 6 × 5.

Thus is a quasi super matrix addition is compatiable if and only if (1) A and B are quasi super matrix of same order and Quasi Super Matrices 185

(2) cell division on A and B are the same that is we have a cell division function Fc: A  B such that Fc(A) = B true.

Now we illustrate this situation by a simple example.

Example 4.22: Let

01234   56789 06284 A =   51739 11012   02101

AA11 12    = AAA21 22 23    A31  where A11, A12, A22, A23 and A31 are sub matrix of A.

10101   01010 11111 B =   11001 10011   01001

BB11 12    =  BB22 23    B31 

We see the total number of columns in B11 and B12 is 5; in B22 and B23 is 5 in B31 and B23 is 5. 186 Neutrosophic Super Matrices and Quasi Super Matrices

Now the addition of A and B is compatible.

01234 10101   56789 01010 06284 11111 A + B =  +   51739 11001 11012 10011   02101 01001

0110 2130   41   5061708190 0161 21 81 41 =  5111703091  11 10 0  0 11  21   0021100011

1133 5   5779 9 1739 5 =   . 627310 2102 3   0310 2

It is important and interesting to note that in case of quasi super matrices even addition is not always compatible even if the matrices are of same order.

Now we proceed on to define multiplication in case of quasi super matrices.

What ever be the situation we need to have compatability of order while multiplying. Quasi Super Matrices 187

Example 4.23: Let

01221   51131 A =   10100   51011  54 and

1011   0102 B = 1010 .   2101   00115x4 be any two quasi super matrices.

1011 01221   0102 51131   AB =  1010 10100   2101 51011 0011

If the partitions or cell divisions are removed imaginarily and the multiplication takes place we get a 4 × 4 matrix.

 4335   12 4 7 11 AB =    2021    7269

188 Neutrosophic Super Matrices and Quasi Super Matrices

The quasi super matrix product assumes an approximate cell division which is not unique for it may not be possible to get a cell division exactly as the very order of it is changed.

We call it as the resultant quasi super matrix. What is more important in this situation is that we can find one or more type of cell division on the product. This is not a problem for the very concept of quasi super matrix was found to apply in fuzzy models.

So to over come all these hurdles we make a mention of the following. As in case of super matrices we first apply the product of a quasi super matrix and its transpose we may recall in case of super matrix we had the product of a super matrix with its transpose gave a nice symmetric super matrix.

We observe that in the case of quasi super matrix A, the transpose AT of A is such that in general the product is not defined. Thus we can define the product only when it is defined.

This is not the case in case of super matrix, for in super matrix we can always define such products but in case of quasi super matrix the product may be defined or not. Only if defined it can be calculated.

This quasi super matrix when its entries are from the fuzzy interval [0,1] we define it to be a fuzzy quasi super matrix or quasi fuzzy super matrix.

All these matrices find their applications when the problem under investigation uses several i.e., one or more fuzzy models simultaneously.

We just illustrate by examples some fuzzy quasi super matrices.

Example 4.24: Let Fs denote the fuzzy quasi super matrix given as

Quasi Super Matrices 189

A1  Fs =  A3  A2 

 0 0.1 0.4 0.5 0.3 0.7    10.11 00.80.5 0.7 0.2 0.5 0.6 1 0.2   = 0.8 0.5 0.6 0.5 0.7 1  , 0.1 0.8 0.6 1 0 0.8   0.8 0.5 0.1 0.9 1 0.2   0.3 0.7 0.5 0.3 0.5 1  where A1 is a 4 × 6 fuzzy rectangular matrix and A2 and A3 are 3 × 3 fuzzy square super matrices.

Example 4.25: Let Ps denote a quasi fuzzy super matrix given by

0.1 1 0.2 0.1 0.9 0.7 0.2 0    0.9 0.1 1 0 0.8 0.6 0.4 0.6 0.7 0.5 0.6 0.8 0.7 0.1 0.5 0.1  0 1 0 0.5 0.4 0.7 0.4 0.3 0.3 1 0.6 0.1 0.5 0 0 0 Ps = ; 0.7 0 0.2 0.4 0.6 0.7 0 0 0.8 1 1 0 1 0 1 1  0.9 0 0.7 0.8 0.9 0.6 0.3 0.2  0.5 0.2 0.6 0.4 0.5 0.1 0.3 0.1 0.6 0.3 0.4 0.3 0.2 0.7 0.1 0  where

AA12   Ps = A ; A 4   3   A5  190 Neutrosophic Super Matrices and Quasi Super Matrices

where A1 and A2 is a 4 × 4 fuzzy matrix. A3 is a 6 × 6 square fuzzy matrix and A4 and A5 are 3 × 2 rectangular fuzzy matrices.

Thus we can have any number of quasi fuzzy super matrices. In case of even quasi fuzzy super matrices we cannot always define the notion of addition or multiplication. Only when special type of compatibility exists we can proceed onto define sum or product or both.

Example 4.26: Let Fs and Ps be two quasi fuzzy super matrices given by

0.3 0.1 0.2 0.7 0.2 0 0.1  1 0.2 0.5 0 0 0.5 1 0 0.3 1 0.1 0.1 0.2 0.3 Fs =  0 0.3 0.8 0.4 1 0.1 0.7 1 0.7 0.3 1 0.1 0.7 1  0.5 0.8 0.9 0 0.3 0.2 0.5

AA12 =   AAA345 and

0 0.7 1 0 0.6 1 0.7  0.3 0 0.8 1 0.5 0.3 1 0.1 0.9 0 0 1 0.2 0 Ps =  0.9 0 0.7 0.5 0.1 0.1 1 0.2 0.3 0.5 0.3 0.5 1 0.7  0.4 0.1 0.6 1 0 0 0.5

BB12 =   BBB345 Quasi Super Matrices 191

We see both min {Ps, Fs}; Ps + Fs = min {Ps, Fs} where min function is well defined.

For if

 12  aaij  ij  Fs =    345 aaaij ij ij 

t where At = ( aij ); t = 1, 2, 3, 4, 5. and  12  bbij  ij  Ps =    345 bijbb ij ij 

p where Bp = ( bij ); p = 1, 2, 3, 4, 5.

11 22 min{aij ,b ij } min{aij ,b ij }  min {Fs + Ps} =  33 44 55 min{aij ,b ij } min{aij ,b ij } max{aij ,b ij }

00.10.200.200.1  0.3 0 0.5 0 0 0.3 1 00.30 00.10.20 =  0 0 0.7 0.4 0.1 0.1 0.7 0.2 0.3 0.3 0.3 0.1 0.7 0.7  0.4 0.1 0.6 0 0 0 0.5

Thus only when compatibility exists we may be in a position to define min function.

Now we proceed onto define max function on Fs and Ps. 192 Neutrosophic Super Matrices and Quasi Super Matrices

11 22 max{aij ,b ij } max{aij ,b ij } max {Fs + Ps} = 33 44 55 max{aij ,b ij } max{aij ,b ij } max{aij ,b ij }

0.30.710.70.610.7  1 0.2 0.8 1 0.5 0.5 1 0.1 0.9 1 0.1 1 0.2 0.3 =. 0.9 0.3 0.8 0.5 1 0.1 1 1 0.7 0.5 1 0.5 1 1  0.5 0.8 0.9 1 0.3 0.2 0.5

Now we proceed onto define max min of Fs, Ps.

i.e., max min {Fs, Ps}

max min{A11 ,B } max min{A2 ,B 2 } =  max min{A33 ,B } max min{A44 ,B } max min{A55 ,B }

0.1 0.3 0.3 0.2 0.6 0.7 0.7   0.2 0.7 1 0.5 0.5 0.2 1  0.3 0.9 0.3 0.3 0.2 0.2 0.3 max min {Fs, Ps} =   0.4 0.3 0.6 1 0.5 0.4 1  0.9 0.3 0.7 0.3 0.5 0.7 0.7   0.5 0.3 0.6 0.3 0 0.2 0.5

At times we may have only ‘addition’ to be compatible. For instance consider the example.

Example 4.27: Let Fs and Ps be any two quasi fuzzy super matrices given by

Quasi Super Matrices 193

0.3 0.8 1 0.4 1 0 0.9  0 0.4 0.5 0.8 0 0.8 0.4 0.8 1 0 0.6 0.7 1 0.3  0.4 0.3 0.8 0.1 0.5 0.6 0.1 Fs = 0.5 0.8 0.6 1 0 0.8 0.1  0.5 1 0 0.5 0.6 1 0 1 0 0.8 1 0.4 0 0.2  0.4 0.5 1 0.3 1 0.4 0.7 and

0.7 0.3 0.5 0.3 0.5 0.1 0  1 0.8 0.4 0.8 0 0.4 0.9 0.7 0.6 0.3 0.1 0.8 0.1 0.3  0.3 0.8 0.5 0.7 0.9 0.7 0.2 Ps = . 0.1 0.5 0.1 0.3 0.4 0.5 0.3  0.30.20.40.70.50.10.8 0.1 0.8 0.3 0.3 0.7 0.2 0.1  0.8 0.7 0.1 0.1 0.6 0.3 0.5

Let us denote

 12 Aa1ij2ij  Aa  Fs =    34 AaAa3ij4ij  and  12 BbBb1ij2ij    Ps =    34 BbBb3ij4ij  max min {Fs, Ps} is not defined as max min {A1, B1} is not defined min or max is defined by

194 Neutrosophic Super Matrices and Quasi Super Matrices

11 22 max(aij ,b ij ) max(aij ,b ij ) max {Fs + Ps} =  33 44 max(aij ,b ij ) max(aij ,b ij )

0.7 0.8 1 0.4 1 0.1 0.9  1 0.8 0.5 0.8 0 0.8 0.9 0.8 1 0.3 0.6 0.8 1 0.3  0.4 0.8 0.8 0.7 0.9 0.7 0.2 = . 0.5 0.8 0.6 1 0.4 0.8 0.3  0.5 1 0.4 0.7 0.6 1 0.8 1 0.8 0.8 1 0.7 0.2 0.2  0.8 0.7 1 0.3 1 0.4 0.7

Thus we see at times max and min may be defined but max min may not be defined.

Here also entries of all these quasi super matrices can be replaced by the entries from neutrosophic ring, dual number ring, dual neutrosophic number rings and so on.

These new structures find applications in constructing the fuzzy neutrosophic quasi super models.

FURTHER READING

1. Birkhoff. G, On the structure of abstract algebras, Proc. Cambridge Philos. Soc, 31, 433-435, 1995.

2. Horst P., Matrix Algebra for Social Scientists, Holt, Rinehart and Winston Inc, 1963.

3. Smarandache. Florentin, An Introduction to Neutrosophy, http://gallup.unm.edu/~smarandache/ introduction.pdf.

4. Smarandache Florentin (editor) Proceedings of the First International Conference on Neutrosophy, Neutrosophic Probability and Statistics, Univ. of New Mexico, Gallup, 2001.

5. Vasantha Kandasamy, W.B. and Florentin Smarandache, Super Linear Algebra, Infolearnquest, Ann Arbor, 2008.

6. Vasantha Kandasamy, W.B. Florentin Smarandache and K. Ilanthenral, Introduction to Bimatrices, Hexis, 2005.

7. Vasantha Kandasamy and Florentin Smarandache, Neutrosophic rings, Hexis, Arizona, U.S.A., 2006.

196 Neutrosophic Super Matrices and Quasi Super Matrices

8. Vasantha Kandasamy, W.B. and Florentin Smarandache, Natural Product n on matrices, Zip Publishing, Ohio, 2012.

9. Vasantha Kandasamy, W.B., Smarandache Rings, American Research Press, Rehoboth, 2002.

10. Vasantha Kandasamy, W.B. and Florentin Smarandache, Finite Neutrosophic Complex Numbers, Zip Publishing, Ohio, 2011.

11. Vasantha Kandasamy, W.B. and Florentin Smarandache, Dual Numbers, Zip Publishing, Ohio, 2012.

12. Vasantha Kandasamy, W.B. and Florentin Smarandache, Special dual like numbers and lattices, Zip Publishing, Ohio, 2012.

13. Vasantha Kandasamy, W.B. and Florentin Smarandache, Special quasi dual numbers and groupoids, Zip Publishing, Ohio, 2012.

INDEX

A

All partitions of a super matrix, 170-2

B

Bilength of a super row bivector, 88-9 Bimatrices, 7-8 Bisuper matrices, 87-9

D

Dual number neutrosophic ring, 7-8

F

Fuzzy quasi super matrix, 188-9

I

Identical in structure, 95-9, 147 198 Neutrosophic Super Matrices and Quasi Super Matrices

M

Major product of super neutrosophic vectors of type A, 35-8 Matrix order of a neutrosophic super matrix, 16-8 Minor product of super neutrosophic vectors of type A, 32-4 Mixed quasi super matrix, 165

N

Natural order of a super neutrosophic matrix, 16-8 Neutrosophic ring, 7-9 Neutrosophic super square matrix, 12 Neutrosophic super vector of type A, 22-5 Neutrosophic super vector of type B, 24-6

Q

Quasi super matrix, 157-9 Quasi super rectangular matrix, 163-5 Quasi super square matrix, 160-2

R

Rectangular super neutrosophic matrix, 14

S

Similar quasi super matrix, 179-2 Simple neutrosophic matrix, 16-8 Special dual like neutrosophic ring, 7-8 Special mixed neutrosophic dual number rings, 7-8 Special quasi dual number neutrosophic ring, 7-8 Super bicolumn vector of type A, 89-91 Super bicolumn vector of type B, 141-3 Super bicolumn vectors, 129 Super bilength of a super column bivector, 91-3 Super bimatrices, 87-9 Super birow vector of type A, 89-91 Index 199

Super birow vectors, 88-9 Super bivectors, 129 Super column bivector of type A, 89-91 Super column neutrosophic matrix, 10 Super neutrosophic rectangular matrix, 14 Super neutrosophic square matrix, 12 Super row bivector of type B, 139-142 Super row bivectors, 88-9 Super matrices, 7-8 Super row neutrosophic matrix, 9-10 Symmetric neutrosophic super matrix, 53-5 Symmetric simple neutrosophic matrix, 45-8

T

Transpose of neutrosophic super vector of type B, 24-6

ABOUT THE AUTHORS

Dr.W.B.Vasantha Kandasamy is an Associate Professor in the Department of Mathematics, Indian Institute of Technology Madras, Chennai. In the past decade she has guided 13 Ph.D. scholars in the different fields of non-associative algebras, algebraic coding theory, transportation theory, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. She has to her credit 646 research papers. She has guided over 68 M.Sc. and M.Tech. projects. She has worked in collaboration projects with the Indian Space Research Organization and with the Tamil Nadu State AIDS Control Society. She is presently working on a research project funded by the Board of Research in Nuclear Sciences, Government of India. This is her 74th book. On India's 60th Independence Day, Dr.Vasantha was conferred the Kalpana Chawla Award for Courage and Daring Enterprise by the State Government of Tamil Nadu in recognition of her sustained fight for social justice in the Indian Institute of Technology (IIT) Madras and for her contribution to mathematics. The award, instituted in the memory of Indian-American astronaut Kalpana Chawla who died aboard Space Shuttle Columbia, carried a cash prize of five lakh rupees (the highest prize-money for any Indian award) and a gold medal. She can be contacted at [email protected] Web Site: http://mat.iitm.ac.in/home/wbv/public_html/ or http://www.vasantha.in

Dr. Florentin Smarandache is a Professor of Mathematics at the University of New Mexico in USA. He published over 75 books and 200 articles and notes in mathematics, physics, philosophy, psychology, rebus, literature. In mathematics his research is in number theory, non-Euclidean geometry, synthetic geometry, algebraic structures, statistics, neutrosophic logic and set (generalizations of fuzzy logic and set respectively), neutrosophic probability (generalization of classical and imprecise probability). Also, small contributions to nuclear and particle physics, information fusion, neutrosophy (a generalization of dialectics), law of sensations and stimuli, etc. He got the 2010 Telesio-Galilei Academy of Science Gold Medal, Adjunct Professor (equivalent to Doctor Honoris Causa) of Beijing Jiaotong University in 2011, and 2011 Romanian Academy Award for Technical Science (the highest in the country). Dr. W. B. Vasantha Kandasamy and Dr. Florentin Smarandache got the 2011 New Mexico Book Award for Algebraic Structures. He can be contacted at [email protected]