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17. MATRICES AND MATRIX TRANSFORMATIONS MATRICES ⎛ 2 1 3 0 ⎞ A matrix is a rectangular array of numbers (or M = ⎜ 5 7 −6 8 ⎟ 3 × 4 symbols) enclosed in brackets either curved or ⎜ ⎟ ⎜ 9 −2 6 −3 ⎟ square. The constituents of a matrix are called entries ⎝ ⎠ or elements. A matrix is usually named by a letter for ⎛ 4 5 7 ⎞ convenience. Some examples are shown below. ⎜ −2 3 6 ⎟ N = ⎜ ⎟ 4 × 3 ⎜ −7 1 0 ⎟ æö314 ⎝⎜ 9 −5 8 ⎠⎟ X = ç÷ èø270- Note that the orders 3 × 4 and 4 × 3 are NOT the æö31426- same. A = ç÷ èø00105- Row matrices ⎡ a b ⎤ F = ⎢ ⎥ ⎣ c d ⎦ If a matrix is composed only of one row, then it is called a row matrix (regardless of its number of ⎡ 4 1 2 ⎤ elements). The matrices �, � and � are row matrices, ⎢ ⎥ Y = ⎢ 3 −1 5 ⎥ ⎢ 6 −2 10 ⎥ J = ()413 L =(30- 42) ⎣ ⎦ K = 21 ⎛ 1 0 ⎞ () I = ⎝⎜ 0 1 ⎠⎟ Column matrices Rows and Columns If a matrix is composed of only one column, then it is called a column matrix (regardless of the number of The elements of a matrix are arranged in rows and elements). The matrices �, � and � are column columns. Elements that are written from left to right matrices. (horizontally) are called rows. Elements that are written from top to bottom (vertically) are called ⎛ ⎞ columns. The first row is called ‘row 1’, the second ⎛ ⎞ 6 4 ⎜ ⎟ ‘row 2’, and so on. The first column is called ⎛ 1 ⎞ ⎜ ⎟ 7 P = ⎜ ⎟ Q = 0 R=⎜ ⎟ ‘column 1, the second ‘column 2’, and so on. ⎝ 2 ⎠ ⎜ ⎟ ⎜ 8 ⎟ ⎝⎜ 3 ⎠⎟ ⎝⎜ 9 ⎠⎟ ⎛ 3 1 4 −1 ⎞ Row 1 ⎜ ⎟ Square matrices M = ⎜ 0 11 −5 8 ⎟ Row 2 ⎜ ⎟ ⎝ 2 1 6 −4 ⎠ Row 3 If a matrix has the same number of rows as the Col 1 Col 2 Col 3 Col 4 number of columns, then it is called square. For example, a matrix that has 6 rows and 6 columns is a ⎛ 3 1 4 −1 ⎞ square matrix. We may describe such a matrix as ⎜ ⎟ being square of order 6 or simply a 66´ matrix. An N = ⎜www.faspassmaths.com0 11 − 5 8 ⎟ mm´ matrix is a square matrix of order m. Q and S ⎜ 2 1 6 − 4 ⎟ ⎝ ⎠ are square matrices. Order of a matrix æö11- æö124 Q = ç÷ ç÷ èø42 S =ç÷--112 The order of a matrix is written as � × �, where m ç÷ represents the number of rows and n represents the 22´ èø0011 number of columns. A matrix of order 4 × 3, consists 33´ of 4 rows and 3 columns while a matrix of order 3 × 4 consists of 3 rows and 4 columns. 160 Copyright © 2019. Some Rights Reserved. www.faspassmaths.com ( The position of elements in a matrix Diagonal matrices Every single entry in a matrix has a specific position A diagonal matrix is a square matrix whose non- that can be uniquely described. In describing the diagonal elements are zero. T and V are diagonal position, we use the notation aij where the subscript, matrices. i, refers to the row number and the subscript j refers to the column number of the element. ⎛ 3 0 0 ⎞ Since each position is unique, no two entries can ⎛ 2 0 ⎞ V = ⎜ 0 −5 0 ⎟ T = ⎜ ⎟ ⎜ ⎟ have the same row and column number. The symbol ⎝ 0 5 ⎠ ⎜ 0 0 7 ⎟ a represents an element. ⎝ ⎠ a belongs to the ith row and the jth column. ij a belongs to the 4th row and the 2nd column. Zero matrix 42 a belongs to the 2nd row and the 4th column. 24 If all the elements of any matrix are zero(s), then the For the matrix, A, defined below, we can assign each matrix is called a zero matrix. Some examples are element its unique position shown below. Notice shown below. A zero matrix can be of any order. a ≠ a . For example, a ≠ a . ij ji 13 31 ⎡ 0 0 ⎤ ⎡ 0 0 0 ⎤ ⎡ 0 0 0 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0 0 0 0 0 ⎛ 4 1 3 2 ⎞ ⎣ ⎦ ⎣ ⎦ ⎜ ⎟ A = ⎜ 0 −11 6 5 ⎟ ⎜ 5 10 2 1 ⎟ ⎝ − − ⎠ Operations on Matrices Col 1 Col 2 Col 3 Col 4 In performing operations on matrices, there are some restrictions. Unlike numbers, one cannot always add, ⎛ 4 1 3 2 ⎞ 11 12 13 14 Row 1 subtract or multiply any two matrices. In fact, a ⎜ ⎟ division of two matrices is not even possible. A = ⎜ 021 −1122 623 524 ⎟ Row 2 ⎜ ⎟ 5 −10 − 2 1 Row 3 ⎝ 31 32 33 34 ⎠ Addition Diagonal elements Matrices can only be added if they are of the same order. This is done by adding or subtracting Diagonal elements are the elements positioned along corresponding entries. The resulting matrix will also the diagonal of a square matrix. A and B are square be of the same order. matrices and have diagonal elements of the type aii . éù12- 1 éù042- A = êú B = êú Square matrix with three diagonal 43- 5 -163 elements ëû ëû 23´ 23´ æö3111 4 ç÷ A = 20 6 éù1211 12-- 1 13 é 011 42 12 13 ù ç÷22 AB+=êú+ê ú ç÷ 43-- 5 1 63 èø--11 5 833 ëû21 22 23ë 21 22 23 û www.faspassmaths.com23´´23 Square matrix with two diagonal éù()()10++() 2-- 4() 12+ elements AB+=êú11 12 13 41+()- ()() 36+ 53 + ëûêú()21 22 23 æö1411 23´ B = ç÷06 èø22 éù121- AB+=êú ëû398 23´ 161 Copyright © 2019. Some Rights Reserved. www.faspassmaths.com Subtraction Solution Matrices can only be subtracted if they are of the æöæö21 6 3 same order. This is done by subtracting (i) 33A ==ç÷ç÷ corresponding entries. The resulting matrix will also èøèø34 912 be of the same order. æöæö412--- 821 éù12- 1 (ii) -22B =-ç÷ç÷= A = èøèø073-- 0146 êú ëû43- 5 23´ éù042- B = Equal matrices êú-163 ëû 23´ Two matrices are equal if: 1. They are of the same order éù1211 12-- 1 13 é 011 42 12 13 ù AB-=êú-ê ú 2. Their corresponding entries are equal. ëû4321 22-- 5 23ë 1 21 63 22 23 û 23´´23 Both conditions must be satisfied before we can deduce that the matrices are equal. Conversely, if éù()()10-----() 2 4() 12 AB-=êú11 12 13 matrices are equal then we can deduce that they must 41--() ()() 36 - 53 - be of the same order and that their corresponding ëûêú()21 22 23 entries are equal. Consider the matrices S and T. They 23´ are of the same order and their corresponding entries éù163- are the same. Therefore, ST= . AB-=êú ëû532- æö13 æö 13 23´ ç÷ ç÷ ST=46- = 46- ç÷ ç÷ ç÷ ç÷ Rule for addition and subtraction of matrices èø20 èø 20 If two matrices have the same order, the matrices Example 2 are said to be conformable to addition and éù42x éù46 subtraction. We obtain the resulting matrix by Given that A = êú B = êú and adding or subtracting corresponding elements. ëû--y 2 ëû22- AB= , find the value of x and of y. Scalar multiplication Solution If AB= , then A and B are of the same order and If a matrix is multiplied by a scalar then each element their corresponding entries are the same. of the matrix is multiplied by the scalar. The resulting Equating corresponding entries: 26x = -y =2 matrix is of the same order. æöabc \x =3 \y =-2 A = ç÷ èødef www.faspassmaths.com æöka kb kc Example 3 kA = ç÷,k is a scalar èøkd ke kf æö12- æö01 Given that A = ç÷ and B = ç÷ and èø13 èø43 Example 1 æö12x æö21 (i) If A = ç÷, then calculate 3� AB-2 =ç÷y , find the value of x, y and z. èø34 ç÷-z èø2 æö412- (ii) If B = ç÷, then calculate −2� èø073- 162 Copyright © 2019. Some Rights Reserved. www.faspassmaths.com Solution We first need to obtain a single matrix for A – 2B. Order of A Order of B Order of C æö12x 2´3 3´2 2´2 AB-2 =ç÷y 3´4 4´2 3´2 ç÷-z èø2 4´2 2´3 4´3 4´3 3´2 4´2 æö12x æöæö12- 01 ç÷ ç÷ç÷-2 =y If two matrices cannot be multiplied, they are said to èøèø13 43ç÷-z èø2 be non-conformable to multiplication. For example, the product YX is not possible since the number of æö12x columns of Y (2) ≠ the number of rows of X, (4) æöæö12- 02ç÷ ç÷ç÷-=y èøèø13 86ç÷-z YX´ÞThis cannot be done. èø2 33´´2 4 æö12x æö14- ç÷ ç÷= y We will now illustrate how two matrices can be èø--73ç÷-z èø2 multiplied using the matrices A and B below. Equating corresponding entries æö1232 -42=x y -3 =-z æö14- 1 ç÷ -7 = A = ç÷ and B =ç÷---1111 24x =- 2 -z =-3 20- 3 èøç÷0462 y èø -4 =-7 z = 3 x = 2 We wish to determine the matrix AB. 2 y =-72 x =-2 () y =-14 First, check to see if this is possible. AB´= C 2424´´33 ´ The Identity or Unit matrix Based on the above rule, the matrix product �� exists If all the diagonal elements of a diagonal matrix are and the product, � will be a 2´4 matrix. equal to one, then it is called the unit or identity matrix and is denoted by U or I only. We write out the structure of the answer, that is, what C looks like. We set up a 2´4 matrix, using the A unit matrix of A unit matrix of order 3 symbol e, to represent an element.
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