MATRICES AND DETERMINANTS MATRICES 1. Matrix :- Arrangement of the elements in rows and column 2. Order of the matrix :- (number of rows) (number of columns) 3. Types Of Matrix 1. Zero matrix :- A matrix in which each element is zero, is called zero matrix or null matrix 2. Row matrix :- A matrix having only one row 3. Column matrix :- A matrix having only one column 4. Square matrix:- A matrix in which the number of rows is equal to the number of columns Principle diagonal: - The diagonal from top left to bottom right 5. Diagonal matrix :- A matrix in which all the elements except the principle diagonal elements are zero 6. Scalar matrix :- A diagonal matrix in which all the principle diagonal elements are equal 7. Unit matrix or Identity matrix :- A diagonal matrix in which each principle diagonal entries is one 8. Idempotent matrix :- A matrix is said to be idempotent if 9. Nilpotent: A square matrix is called nilpotent matrix if there exist a positive integer ‘n’ such that . If ‘m’ is the least positive integer such that , them ‘m’ is called the index of the nilpotent matrix. 10. Involutory matrix :- A matrix is said to be involutory if 11. Upper triangular matrix :- A matrix in which all the elements below the principle diagonal are zero 12. Lower triangular matrix :- A matrix in which all the elements above the principle diagonal are zero 4. Trace of a matrix: - Trace of a matrix is defined and denoted as ( ) sum of principal diagonal element. 5. Two matrices are said to be equal if they have the same order and the corresponding elements are also equal.
6. If ( ) ( ) be two matrices of order then
1. ( ) [scalar (k) multiplication of matrix]
2. ( ) [matrix addition]
3. ( ) [matrix subtraction]
7. Matrix multiplication :- If ( )be the matrix of order and ( )be the matrix of order
then ( ) be the matrix of order , where is the sum of the product of the corresponding elements of row of A and column of B. 8. If are matrices then 1) ) need not be equal to ) 9. Transpose of a matrix :- A matrix obtained from interchanging the rows into columns. If is the matrix
then transpose of A is denoted by or i.e., ( ) then ( ) 10. Orthogonal matrix :- A matrix is said to be orthogonal if 11. Symmetric matrix :- A matrix is said to be symmetric if 12. Skew-symmetric matrix :- A matrix is said to be skew-symmetric if (diagonal elements of skew-symmetric matrices are zero)
KCET SPSM PU SCIENCE COLLEGE 1 MATRICES AND DETERMINANTS 13. If is a square matrix , then 1) is a symmetric matrix 2) is a Skew-symmetric matrix ) is a symmetric matrix 14. If are square matrix then is symmetric iff 15. Properties of matrices :- If and are three matrices then 1. ( ) 2. ( ) 3. ( ) 4. ( ) 5. ( ) Where 6. ( ) ( ) i.e., matrix multiplication is associative 7. is need not be true. i.e., matrix multiplication is not commutative DETERMINANTS
16. If ( ) is a square matrix of order 2 then (determinate of )
| | | |
17. If ( ) then | | ( ) ( )
( ) 18. Properties Of Determinants 1. The value of a determinant is unaltered if its rows and columns are interchanged ( | | | |) 2. If two rows (or columns) of a determinant are interchanged then sign of the determinant changes 3. If in a determinant two rows (or columns) are identical then the value of the determinant is zero 4. If the element of any row (or column) are multiplied by k then the value of the determinant is multiplied by k Note: - | | | | where is a constant and is the order of the matrix 5. If in a determinant one row (or column) is multiple of another then the value of the determinant is zero 6. If each element in any row (or column) of a determinant is a sum of two terms then the determinant can be expressed as a sum of 2 determinants 7. If to the elements of any row (or column) of a determinant the same multiples of the corresponding elements of the other rows (or columns) of the determinant are added then the value of the determinant is unaltered 8. If and are the square matrices of same order then | | | || | 9. The value of a determinant of upper triangular matrix or lower triangular matrix is equal to product of the principal diagonal element. 10. Determinant of a skew symmetric matrix of odd order is zero and of even order is a non-zero perfect square
19. The minor of an element is the value of the determinant obtained by deleting row and
column of the matrix. The minor of the element is denoted by
20. The cofactor of an element of a matrix is denoted and defined by ( ) 21. The cofactor matrix of a square matrix is the matrix obtained by replacing the elements of by its corresponding cofactors 22. The adjoint of the matrix is the transpose of the cofactor matrix and it is denoted by 23. If is any square matrix then ( ) ( ) | |
KCET SPSM PU SCIENCE COLLEGE 2 MATRICES AND DETERMINANTS 24. If A is any square matrix and | | then | | | | where n is the order of the matrix 25. If | | then A is called singular matrix and If | | then A is called non-singular matrix 26. If and are square matrix and is non-singular then
27. If is non-singular then (inverse of ) ( ) | |
28. If ( ) then ( )
29. If is non-singular then ( ) ( ) 30. If is non-singular then ( ) ( ) ( )
31. If is non-singular then | | | | 32. Solving the system of linear equations
, and
1) Cramers Rule :- If | | , | | ,
| | and | | then unique solution : If and solution is ,
and
2) Matrix method :- If [ ] [ ] and [ ] then unique solution : If
and solution is 33. Note: 1) If | | then there exist unique solution and the system of equation is consistent. 2) If | | and ( ) , then system of equation has infinitely many solutions and the system of equation is consistent. 3) If | | and ( ) , then system of equation has no solutions and the system of equation is inconsistent. 34. If is a square matrix and is the identity matrix of same order then is called characteristic matrix 35. | | is called characteristic equation of A and its roots are called characteristic roots or Eigen values 36. The Eigen values of upper or lower triangular matrix are the principal diagonal elements. 37. If is a square matrix of order then characteristic equation is given by ( ) | | , Where ( ) is a trace of the matrix A. Note:- 1. If A is a square matrix of order n, then A will have at most n Eigen values 2. If A is a square matrix then A and A’ have the same Eigen values 3. A is singular if and only if 0 is an Eigen value of A. 4. Sum of the Eigen values is always equal to the trace of the matrix.
5. If are the Eigen values of then are the Eigen values of
38. Cayley-Hamilton theorem :- Every square matrices satisfies its characteristic equations
KCET SPSM PU SCIENCE COLLEGE 3 MATRICES AND DETERMINANTS PROBLEMS 1) If A is a square matrix then A AT is 1) Symmetric 2) Skew Symmetric 3) A Scalar Matrix 4) A unit Matrix 2) If are matrices of same order then ( ) is 1) skew symmetric matrix 2) null matrix 3) symmetric matrix 4) unit matrix 3) The matrix A is both symmetric and skew symmetric, then 1) A is diagonal matrix 2) A is zero matrix 3) A is a square matrix 4) none of these 4) Matrices A and B will be inverse of each other only if 1) 2) 3) 4) 5) If P, Y and W are the matrices of order respectively, such that is defined, then 1) 2) 3) 4) 6) The total number of possible matrices of order 3 X 3 with each entry 2 or 0 is 1) 2) 3) 4)
7) If the matrix [ ] is skew symmetric matrix, then
1) 4 2) 0 3) 4) 10
8) The symmetric part of the matrix ( )
1) ( ) 2) ( ) 3) ( ) 4) ( )
9) If ( ) then if the value of is
1) 2) 3) 4)
10) If [ ] [ ] [ ] then the values of are respectively.
1) 1, 0, 1 2) 1, 1, 0 3) 0, 1, 1 4) 1, 1, 1
11) If ( ) ( ) then the value of
1) 2) 3) 4) 12) If A is an orthogonal matrix then 1) | | 2) | | 3) | | 4) none of these 13) If A = diag(a,b,c) then An is 1) abc 2) diag (na, nb, nc) 3) diag (anbncn) 4) anbncn
14) If ( ) then
1) ( ) 2) ( ) 3) ( ) 4) ( ) 15) If A is a square matrix such that then ( ) 1) 2) 3) 4) 16) If A is a square matrix such that then, ( ) ( ) 1) 2) 3) 4)
KCET SPSM PU SCIENCE COLLEGE 4 MATRICES AND DETERMINANTS
17) On using elementary column operations in the following matrix equation ( )
( ) ( ) we have,
1) ( ) ( ) ( ) 2) ( ) ( ) ( )
3) ( ) ( ) ( ) 4) ( ) ( ) ( )
18) If A,B are square matrices of order 3 X 3 such that |A| = -1, |B| = 3, then | | 1) 2) 3) 4) 19) If is singular then is 1) Singular 2) non-singular 3) symmetric 4) skew-symmetric 20) If and then 1) 2) 3) 4) 21) If are two matrices such that then 1) 2 2) 3) 4)
22) If [ ] then ( )
1) ( ) 2) ( ) 3) ( ) 4) ( )
23) If is a matrix of order such that ( ) then | | 1) 30 2) 40 3) 100 4) 10
24) If is the root of the equation then the value of | | is
1) 0 2) 1 3) 4)
25) If , , are the roots of = 0, then the value of the determinant | | is
1) 2) 3) 4) 26) The value of the determinant of a skew symmetric matrix of odd order is 1) Zero 2) Perfect square 3) Can’t be predicted 4) none of these
27) There are two values of a which makes determinant | | then sum of these
numbers is 1)4 2) 5 3) -4 4) 9
28) If | | | |
1) 2) 2k 3) 3k 4) 6k
29) If | | then ( )
1) ( ) 2) ( ) 3) ( ) 4) ( )
KCET SPSM PU SCIENCE COLLEGE 5 MATRICES AND DETERMINANTS
30) The value of | | =
1) abc 2) a + b + c 3) 0 4) 1
31) In triangle |( ) ( ) ( ) | , then the triangle is ( ) ( ) ( ) 1) Equilateral 2) Isosceles 3) Right angled 4) Scalene
32) The value of | |
1) ( )( )( ) 2)( )( )( ) 3) 4) 0
33) If ax4 + bx3 + cx2 + dx + e = | | then e =
1) 2 2) 5 3) 0 4) 1
34) The value of | |
1) ( ) 2) ( ) 3) ( ) 4) ( )
35) The value of | |
1) ( )( )( ) 2) ( )( )( ) 3) ( )( )( ) 4) ( )( )( )
36) The value of | |
1) ( ) 2) ( ) ( ) ( ) 3) ( ) ( ) ( ) 4) 1 ( ) 37) If then the value of | ( ) | ( ) 1) 2) 3) 4) none of these
38) When are in GP the value of | |
1) 2) 3) 4)
39) If are all different from zero and | | then the value of
1) 2) 3) 4)
KCET SPSM PU SCIENCE COLLEGE 6 MATRICES AND DETERMINANTS
40) If | | then the value of is
1) 2 2) 1 3) 0 4) pqr ( ) ( )
41) If | | ∑
1) 2) 3) 4)
42) The value of | | is
1) 2) 1 3) 0 4) none of these
43) If = | | | |
1) = 2 2) = 2 3) = 4) =4
44) The root of a equation | |
1) a 2) b 3) 0 4) 1
45) If | | and is the cofactor of , then the value of is given by
1) 2)
3) 4)
46) The inverse of the matrix [ ] is
1) [ ] 2) [ ]
3) 4) [ ]
[ ]
47) The inverse of the matrix[ ] is
1) [ ] 2) [ ] 3) [ ] 4) [ ]
48) The inverse of ( ) is
1) ( ) 2) ( )
3) ( ) 4) ( )
KCET SPSM PU SCIENCE COLLEGE 7 MATRICES AND DETERMINANTS
49) Let X = [ ] ; D = [ ] & A = [ ] .If X = A-1 D, then X is equal to
1) [ ] 2) [ ] 3)[ ] 4) [ ]
50) The value of for which the system of equations & does not have a solution is 1) 3 2) 3) 0 4) 1 51) If the system of linear equations and has non-zero solution, then are in 1) A.P 2) G.P 3) H.P 4) None of these 52) The number of values of for which the lines and possesses a non-zero solution is ) 2) 1 3) 2 4) 3 53) The system of equations has 1) Infinitely many solutions 2) A unique solution 3) No solutions 4) Finitely many solutions
54) The number of distinct real roots of | | in
1) 2) 3) 4)
55) Let ( )
1) ( ) 2) ( ) ( ) 3) ( ) ( ) 4) ( ) [ ]
( ) 56) Let ( ) | | then
1) 2) 3) 4) Home Work
1. If A = [ ]
1) [ ] 2) [ ] 3) [ ] 4) [ ]
2. If 2A + 3B = [ ] [ ]
1) [ ] 2) [ ] 3) [ ] 4) [ ]
3. If A = [ ]
1) 2 2) 3 3) 7 4) 5
4. If [ ]
1) 3 2) 5 3) 4 4) 2
5. If A = [ ]
1) 0, 1, -1 2) 0, + 12, -12 3) 0, 5, -5 4) 0, 4, -4
KCET SPSM PU SCIENCE COLLEGE 8 MATRICES AND DETERMINANTS
6. The solutions of the equation | |
1) 3, -1 2) -3, 1 3) 3, 1 4) -3, -1
7. If x = -9 is a root of the equation | | =0 then other two roots are
1) 1, 5 2) 2, -7 3) -2, 7 4) 2,7
8. If 5 is one root of the equation | | then the other two roots of
the equation are 1) -2, 7 2) -2, -7 3) 2,7 4)
9. If | |
1) 2) 3) 4)
10. If A + B = [ ] and [ ]
1) [ ] 2) [ ] 3) [ ] 4) none of these
11. If n is a non-negative integer and A = [ ]
1) [ ] 2) [ ] 3) [ ] 4) [ ]
12. If A = [ ]
1) [ ] 2) [ ] 3) [ ] 4) [ ]
13. If in a square matrix A = (aij), we find that aij = aji for all i, j, then A is 1) Symmetric matrix 2) triangular matrix 3) transpose matrix 4) skew symmetric 14. Choose the correct answer 1) Every scalar matrix is an identity matrix 2) Every identity matrix is a scalar matrix 3) Every diagonal matrix is an identity matrix 4) A square matrix whose each element is 1 is an identity matrix
15. The matrix [ ]
) 2) 3) 4) skew symmetric matrix
16. If A = [ ]
1) Does not exist 2) [ ] 3) [ ] 4) [ ]
17. If A and B are square matrices and ( ) 1) 2) 3) 4)
18. For how many values of x in the interval [ ] the matrix [ ] is singular?
1) 2 2) 0 3) 3 4) 1
KCET SPSM PU SCIENCE COLLEGE 9 MATRICES AND DETERMINANTS 19. Assuming that the sums and products given below are defined, which of the following is not true for matrices? ) 2) 3) ( ) 4)
20. [ ] [ ]
1) [ ] 2) [ ] 3) [ ] 4) [ ]
21. [ ] [ ] | |
1) 80 2) 3) 4)
22. | |
1) (x+p) (x+q) (x – p – q) 2) (x - p) (x - q) (x + p + q) 3) (x + p) (x - q) (x + p + q) 4) (x+p)(x+q)(x+p+q)
23. | |
1) 41 2) 51 3) 31 4)
24. | |
1) 2) √ 3) √ 4) √ √
25. | |
) ( )( )( ) 2) 3) ( )( )( ) 4) ( )( )( )
26. | |
1) 2) 36 3) 12 4) 0
27. | |
1) 0 2) abc 3) 4 abc 4) | |
28. The value of | |
1) 8 abcd 2) abcd 3) 4 abcd 4) 6 abcd
29. The value of | |
1) 0 2) a + b + c 3) 4 abc 4)
KCET SPSM PU SCIENCE COLLEGE 10 MATRICES AND DETERMINANTS
30. If a, b, c are non real numbers, then the value of A = | |
1) 0 2) ab + bc + ca 3) abc 4)
31. The value of | |
1) 1 2) 0 3) 2 4) 4
32. If √ | |
1) 3 2) 0 3) 1 4) -1
33. The value of | |
1) 1 2) 0 3) (a - b) (b – c) (c – a) 4) (a + b) (b + c) (c + a)
34. The value of | |
1) 441 x 446 x 451 2) 0 3) -1 4) 1
35. The value of | |
1) ( ) 2) ( ) 3) 0 4) ( )
36. The value of | |
1) 2) 3) 4)
37. | |
1) 2) 3) 2 4) 3
38. | |
1) 1 + a + b + c 2) 1 + ab + bc + ca 3) 1+ a2 + b2 + c2 4) abc
39. | |
1) (x – p) (x – q) ( x + p + q) 2) ( )( ) 3) ( )( ) 4) ( p – q) ( x – q) ( x –p)
40. If , , are the roots of x3 + px + q = 0, then the value of the determinant | | is
1) 2) 3) 4)
41. If ( ) | ( ) ( ) | ( ) ( ) ( )( ) ( ) ( ) 1) 0 2) 1 3) 200 4) -200
KCET SPSM PU SCIENCE COLLEGE 11 MATRICES AND DETERMINANTS
42. If | |
1) 2) 3) 4)
43. If A = | | | |
1) 9 2) 1/9 3) 91 4) 81 44. If A and B are square matrices of the same order such that (A + B) (A – B) = A2- B2 then (ABA-1)2 1) 1 2) A2B2 3) A2 4) B2
45. If A = | | | |
1) B = A2 2) B = 0 3) A = B 4)
46. The inverse of the matrix [ ]
) [ ] 2) [ ]
3) [ ] 4) [ ]
47. [ ]
1) [ ] 2) [ ] 3) [ ] 4) [ ]
48. [ ] is
1) [ ] 2) [ ]
3) [ ] 4) [ ]
49. [ ]
1) [ ] 2) [ ] 3) [ ] 4) [ ]
50. If A = [ ] [ ] and then =
1) 2) 2 3) 4)
51. If A = [ ] ( )
1) [ ] 2) [ ] 3) [ ] 4) [ ]
52. If A = [ ]
1) [ ] 2) [ ] 3) [ ] 4) [ ]
53. If [ ] [ ] [ ]
1) 2) 3) 4)
KCET SPSM PU SCIENCE COLLEGE 12 MATRICES AND DETERMINANTS
54. If A = [ ]
1) -4 2) 0 3) 1 or 4 4) 4 and not 1 55. If A is a square matrix of order 3 and |A| = 8, then |adj A| =
1) 8 2) 82 3) 83 4)
56. The characteristic roots of the matrix [ ]
1) 2, 4,6 2) 3) 4) 1, 3, 6
57. If ax4 + bx3 + cx2 + dx + e = | | then e =
1) 2 2) 5 3) 0 4) 1
58. The symmetric part of the matrix ( )
1) ( ) 2) ( ) 3) ( ) 4) ( )
59. If is a matrix of order such that ( ) then | | 1) 30 2) 40 3) 100 4) 10
60. The inverse of the matrix [ ] is
1) 2) [ ] 3) 4)
( ) ( ) ( )
ANSWER KEY
HOME WORK Q A Q A Q A Q A Q A Q A Q A Q A Q A Q A 1 3 2 2 3 1 4 2 5 2 6 1 7 4 8 4 9 1 10 3 11 1 12 3 13 1 14 2 15 4 16 1 17 1 18 4 19 4 20 1 21 2 22 2 23 1 24 2 25 1 26 4 27 1 28 1 29 1 30 1 31 2 32 2 33 3 34 2 35 3 36 4 37 1 38 3 39 1 40 2 41 1 42 3 43 4 44 4 45 3 46 4 47 2 48 4 49 1 50 3 51 4 52 4 53 2 54 4 55 2 56 4 57 3 58 1 59 3 60 1
KCET SPSM PU SCIENCE COLLEGE 13 MATRICES AND DETERMINANTS TEST
1) If A = ( ) ( ) and A + 2x = B, then x =
1) ( ) 2) ( ) 3) ( ) 4) ( )
2) If order of matrix A = 4 x 3, order of matrix B = 4 x 5 and order of matrix C = 7 x 3 then the order of (A| x B)| x C| is 1) 4 x 5 2) 3 x 7 3) 4 x 3 4) 5 x 7 3) If trace of the matrix is then the trace of the matrix is
1) 2) 3) 8 4) 15
4) Let A = [ ] then An is
1) [ ] 2) [ ] 3) [ ] 4) [ ]
5) If [ ] [ ] then | |
1) 80 2) 100 3) 100 4) 80
6) If 1, , 2 are the cube roots of unity, then | | has value
1) 0 2) 1 3) 4) 2
7) | | is equal to
1) 2) 0 3) 4) none of these 8) Assuming that the sum and the products given below are defined, which of the following is not true for the matrix? 1) 2) ( ) 3) 4)
9) The roots of the equation | | are
1) (0, 9, 16) 2) (0, 12, 12) 3) (0, 12, -12) 4) (0, 12, 16) 10) If A is a square matrix of a order 3 and |A| =3, then |adjA| is 1) 3 2) 9 3) 1/3 4) 0 11) The value of the third order determinant is 11 then the value of the square of the determinant formed by the cofactors will be 1) 11 2) 121 3) 1331 4) 14641
12) The adjoint of the matrix ( )
1) 2) 3) 4)
13) The inverse of the matrix ( ) is
1) ( ) 2) ( ) 3) ( ) 4) ( )
14) If AB A, BA B and I) ABA22 II) ABA A III) AA2 , BB2 Then which of the above statement is/are correct 1) All the above I, II, III 2) Only I and II 3) Only II and III 4) Only I and III
KCET SPSM PU SCIENCE COLLEGE 14 MATRICES AND DETERMINANTS
15) If A = [ ]is a symmetric then is
1) 3 2) 5 3) 2 4) 4
16) If [ ] [ ] = [ ] then
1) a=1, b=1 2) , b = 3) a = , b = 4) , b = 17) The value of for which the system of equations ( ) ( ) , ( ) ( ) and has no solution is 1) 1 2) 3) 4) 2
18) If are non zero numbers then the inverse of the matrix ( )
1) ( ) 2) ( )
3) ( ) 4) ( )
19) If A and B are symmetric matrices of same order, then 1) skew symmetric matrix 2) Symmetric matrix 3) zero matrix 4) Identity matrix
20) | |
1) 2) 3) 4)
21) On using elementary row operation in the following matrix equation
( ) ( ) ( ) we have,
1) ( ) ( ) ( ) 2)( ) ( ) ( )
3) ( ) ( ) ( ) 4) ( ) ( ) ( )
22) If ( )
1) ( ) 2) ( ) 3) ( ) 4) ( )
23) ( ) is such that then 1) 2) 3) 4)
24) | |
1) 2) 3) 4) 25) The area of the triangle with vertices ( ) ( ) ( ) is 9 sq. units. The value of k is 1) 2) 3) 4)
26) The value of the determinant | |
1) ( ) 2) ( ) 3) ( ) 4) ( )
KCET SPSM PU SCIENCE COLLEGE 15 MATRICES AND DETERMINANTS
27) If ( ) then
1) 2) 3) 4)
28) If ( ) | | then,
1) ( ) 2) ( ) 3) ( ) 4) ( )
29) The maximum value of | | ( )
√ √ 1) 2) 3) √ 4)
30) If A, B and C are angles of a triangle then the determinant | |
1) 2) 3) 4) 31) Let A be a square matrix of order 3 X 3, then | | is equal to 1) | | 2) | | 3) | | 4) | | 32) Which of the following is correct 1) determinant is a square matrix 2) determinant is a number associated to a matrix 3) determinant is a number associated to a square matrix 4) none of these 33) If area of triangle is 35 sq. units with vertices ( ) ( ) ( ) 1) 2) 3) 4) 34) If A is an invertible matrix of order 2 X 2, then ( ) is equal to
1) 2) 3) 4)
KCET SPSM PU SCIENCE COLLEGE 16