Q: If M_k= bmatrix k & k-1 \\ k-1 & k bmatrix where k is a natural number, then what is |M_1|+|M_2|+|M_3|+ +|M_ 50 | equal to ?

Given M_k = bmatrix k & k-1 \\ k-1 & k bmatrix The determinant of M_k is given by: |M_k| = k^2 - (k-1)^2 |M_k| = k^2 - (k^2 - 2k + 1) = 2k - 1 The required sum is: |M_1| + |M_2| + |M_3| + + |M_ 50 | = _ k=1 ^ 50 (2k - 1) This represents the sum of the first 50 odd natural numbers, which forms an arithmetic progression: 1 + 3 + 5 + + 99 The sum of an arithmetic progression is given by n 2 (a + l), where n = 50, a = 1, and l = 99: Sum = 50 2 (1 + 99) = 25 100 = 2500 Answer: 2500

Q: For the following two (02) items: Let A = pmatrix & \\ - & pmatrix . What is [ adj \, A]^ -1 equal to?

Given A = pmatrix & \\ - & pmatrix The determinant of A is |A| = ^2 - (- )( ) = ^2 + ^2 = 1 For any non-singular matrix, the inverse is given by A^ -1 = 1 |A| adj \, A Substituting |A| = 1, we get adj \, A = A^ -1 Taking the inverse on both sides yields [ adj \, A]^ -1 = (A^ -1 )^ -1 = A Answer: A

Q: Consider the following statements about the matrix M = pmatrix 71 & 23 & 48 \\ 57 & 28 & 29 \\ 65 & 17 & 48 pmatrix : Statement-I: The inverse of M does not exist. Statement-II: M is non-singular. Whi

The given matrix is M = pmatrix 71 & 23 & 48 \\ 57 & 28 & 29 \\ 65 & 17 & 48 pmatrix . Let C_1, C_2, C_3 be the first, second, and third columns of the matrix M respectively. Observing the elements of the columns, we get: C_2 + C_3 = pmatrix 23 + 48 \\ 28 + 29 \\ 17 + 48 pmatrix = pmatrix 71 \\ 57 \\ 65 pmatrix = C_1 Since the first column is the sum of the second and third columns, the columns of M are linearly dependent. Applying the column operation C_1 C_1 - (C_2 + C_3), the first column bec

Q: If [x \ 1 \ 1] pmatrix 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 pmatrix pmatrix 1 \\ 1 \\ x pmatrix = [45], then which one of the following is a value of x?

Given equation is [x \ 1 \ 1] pmatrix 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 pmatrix pmatrix 1 \\ 1 \\ x pmatrix = [45] Multiplying the first two matrices: [x(1) + 1(4) + 1(7) x(2) + 1(5) + 1(8) x(3) + 1(6) + 1(9)] pmatrix 1 \\ 1 \\ x pmatrix = [45] [x + 11 2x + 13 3x + 15] pmatrix 1 \\ 1 \\ x pmatrix = [45] Multiplying the resulting row matrix with the column matrix: (x + 11)(1) + (2x + 13)(1) + (3x + 15)(x) = 45 x + 11 + 2x + 13 + 3x^2 + 15x = 45 3x^2 + 18x + 24 = 45 3x^2 + 18x - 21 = 0 x^2 + 6x

Q: If A = pmatrix 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 pmatrix , then what is A^2 - 4A equal to?

Given A = pmatrix 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 pmatrix First, calculating A^2: A^2 = pmatrix 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 pmatrix pmatrix 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 pmatrix A^2 = pmatrix 1+4+4 & 2+2+4 & 2+4+2 \\ 2+2+4 & 4+1+4 & 4+2+2 \\ 2+4+2 & 4+2+2 & 4+4+1 pmatrix = pmatrix 9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9 pmatrix Now, calculating 4A: 4A = 4 pmatrix 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 pmatrix = pmatrix 4 & 8 & 8 \\ 8 & 4 & 8 \\ 8 & 8 & 4 pmatrix Subtracting 4A from A^

Question 1

Consider the following statements : I. If n n (n>1) matrix is symmetric, then its inverse is also a symmetric matrix. II. If n n (n>1) matrix is singular, then its adjoint is also a singular matrix. W

Accepted Answer

Let A be an invertible symmetric matrix of order n n. Then A^T = A. We know that (A^ -1 )^T = (A^T)^ -1 . Substituting A^T = A, we get (A^ -1 )^T = A^ -1 . Thus, the inverse of a symmetric matrix is also symmetric. The first statement is correct. Let A be a singular matrix of order n n. Then |A| = 0. The determinant of the adjoint of A is given by | adj (A)| = |A|^ n-1 . Since n > 1 and |A| = 0, we have | adj (A)| = 0^ n-1 = 0. Thus, the adjoint of a singular matrix is also a singular matrix. Th

Question 2

If M is a square matrix such that M^3=M, then how many values of |M| are possible ?

Accepted Answer

Given M^3 = M Taking determinant on both sides, we get: |M^3| = |M| Using the property |A^n| = |A|^n, we have: |M|^3 = |M| |M|^3 - |M| = 0 |M|(|M|^2 - 1) = 0 |M|(|M| - 1)(|M| + 1) = 0 |M| = 0, 1, -1 Thus, there are three possible values for |M|. Answer: Three

Question 3

If M_k= bmatrix k & k-1 \ k-1 & k bmatrix where k is a natural number, then what is |M_1|+|M_2|+|M_3|+ +|M_ 50 | equal to ?

Accepted Answer

Given M_k = bmatrix k & k-1 \ k-1 & k bmatrix The determinant of M_k is given by: |M_k| = k^2 - (k-1)^2 |M_k| = k^2 - (k^2 - 2k + 1) = 2k - 1 The required sum is: |M_1| + |M_2| + |M_3| + + |M_ 50 | = _ k=1 ^ 50 (2k - 1) This represents the sum of the first 50 odd natural numbers, which forms an arithmetic progression: 1 + 3 + 5 + + 99 The sum of an arithmetic progression is given by n 2 (a + l), where n = 50, a = 1, and l = 99: Sum = 50 2 (1 + 99) = 25 100 = 2500 Answer: 2500

Question 4

If M^ T is the transpose of a 2 2 matrix M, then which of the following is/are correct ? I. |M+M^ T |=|M|+|M^ T | if M is symmetric. II. |M+M^ T |=0 if M is anti-symmetric. Select the answer using the

Accepted Answer

For Statement I: If M is a symmetric matrix of order 2 2, then M^ T = M. |M+M^ T | = |2M| = 2^2|M| = 4|M| |M| + |M^ T | = |M| + |M| = 2|M| Since 4|M| 2|M| for all symmetric matrices, Statement I is incorrect. For Statement II: If M is an anti-symmetric matrix, then M^ T = -M. M+M^ T = M - M = O |M+M^ T | = |O| = 0 Statement II is correct. Answer: II only

Question 5

Consider the following in respect of non-singular matrices A and B: I. (AB)^ -1 = A^ -1 B^ -1 II. (BA)(AB)^ -1 = I, where I is the identity matrix III. (AB)^T = A^T B^T How many of the above are corre

Accepted Answer

For Statement I, the reversal law of matrix inverse gives (AB)^ -1 = B^ -1 A^ -1 . Since matrix multiplication is not commutative in general, B^ -1 A^ -1 A^ -1 B^ -1 . Thus, Statement I is incorrect. For Statement II, using the reversal law, (BA)(AB)^ -1 = (BA)(B^ -1 A^ -1 ) = B(AB^ -1 )A^ -1 = BIA^ -1 = BA^ -1 . This is not equal to I in general. Thus, Statement II is incorrect. For Statement III, the reversal law of matrix transpose gives (AB)^T = B^T A^T. Since matrix multiplication is not co

Question 6

For the following two (02) items: Let A = pmatrix & \ - & pmatrix . What is [ adj \, A]^ -1 equal to?

Accepted Answer

Given A = pmatrix & \ - & pmatrix The determinant of A is |A| = ^2 - (- )( ) = ^2 + ^2 = 1 For any non-singular matrix, the inverse is given by A^ -1 = 1 |A| adj \, A Substituting |A| = 1, we get adj \, A = A^ -1 Taking the inverse on both sides yields [ adj \, A]^ -1 = (A^ -1 )^ -1 = A Answer: A

Question 7

Consider the following statements about the matrix M = pmatrix 71 & 23 & 48 \ 57 & 28 & 29 \ 65 & 17 & 48 pmatrix : Statement-I: The inverse of M does not exist. Statement-II: M is non-singular. Whi

Accepted Answer

The given matrix is M = pmatrix 71 & 23 & 48 \ 57 & 28 & 29 \ 65 & 17 & 48 pmatrix . Let C_1, C_2, C_3 be the first, second, and third columns of the matrix M respectively. Observing the elements of the columns, we get: C_2 + C_3 = pmatrix 23 + 48 \ 28 + 29 \ 17 + 48 pmatrix = pmatrix 71 \ 57 \ 65 pmatrix = C_1 Since the first column is the sum of the second and third columns, the columns of M are linearly dependent. Applying the column operation C_1 C_1 - (C_2 + C_3), the first column bec

Question 8

If [x \ 1 \ 1] pmatrix 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 pmatrix pmatrix 1 \ 1 \ x pmatrix = [45], then which one of the following is a value of x?

Accepted Answer

Given equation is [x \ 1 \ 1] pmatrix 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 pmatrix pmatrix 1 \ 1 \ x pmatrix = [45] Multiplying the first two matrices: [x(1) + 1(4) + 1(7) x(2) + 1(5) + 1(8) x(3) + 1(6) + 1(9)] pmatrix 1 \ 1 \ x pmatrix = [45] [x + 11 2x + 13 3x + 15] pmatrix 1 \ 1 \ x pmatrix = [45] Multiplying the resulting row matrix with the column matrix: (x + 11)(1) + (2x + 13)(1) + (3x + 15)(x) = 45 x + 11 + 2x + 13 + 3x^2 + 15x = 45 3x^2 + 18x + 24 = 45 3x^2 + 18x - 21 = 0 x^2 + 6x

Question 9

If A = pmatrix 1 & 2 & 2 \ 2 & 1 & 2 \ 2 & 2 & 1 pmatrix , then what is A^2 - 4A equal to?

Accepted Answer

Given A = pmatrix 1 & 2 & 2 \ 2 & 1 & 2 \ 2 & 2 & 1 pmatrix First, calculating A^2: A^2 = pmatrix 1 & 2 & 2 \ 2 & 1 & 2 \ 2 & 2 & 1 pmatrix pmatrix 1 & 2 & 2 \ 2 & 1 & 2 \ 2 & 2 & 1 pmatrix A^2 = pmatrix 1+4+4 & 2+2+4 & 2+4+2 \ 2+2+4 & 4+1+4 & 4+2+2 \ 2+4+2 & 4+2+2 & 4+4+1 pmatrix = pmatrix 9 & 8 & 8 \ 8 & 9 & 8 \ 8 & 8 & 9 pmatrix Now, calculating 4A: 4A = 4 pmatrix 1 & 2 & 2 \ 2 & 1 & 2 \ 2 & 2 & 1 pmatrix = pmatrix 4 & 8 & 8 \ 8 & 4 & 8 \ 8 & 8 & 4 pmatrix Subtracting 4A from A^

Question 10

If A = pmatrix y & z & x \ z & x & y \ x & y & z pmatrix where x, y, z are integers, is an orthogonal matrix, then what is the value of x^2 + y^2 + z^2?

Accepted Answer

For an orthogonal matrix A, the condition A A^T = I must be satisfied. Given A = pmatrix y & z & x \ z & x & y \ x & y & z pmatrix , we find A^T = pmatrix y & z & x \ z & x & y \ x & y & z pmatrix . Multiplying A and A^T gives: A A^T = pmatrix y & z & x \ z & x & y \ x & y & z pmatrix pmatrix y & z & x \ z & x & y \ x & y & z pmatrix = pmatrix x^2+y^2+z^2 & xy+yz+zx & xy+yz+zx \ xy+yz+zx & x^2+y^2+z^2 & xy+yz+zx \ xy+yz+zx & xy+yz+zx & x^2+y^2+z^2 pmatrix Equating A A^T to the identity

Question 11

If A = pmatrix x & y & z \ y & z & x \ z & x & y pmatrix where x, y, z are integers, is an orthogonal matrix, then what is A^2 equal to?

Accepted Answer

Given the matrix A = pmatrix x & y & z \ y & z & x \ z & x & y pmatrix . The transpose of A is given by: A^T = pmatrix x & y & z \ y & z & x \ z & x & y pmatrix We can observe that A^T = A, which means A is a symmetric matrix. It is given that A is an orthogonal matrix. By the definition of an orthogonal matrix, we have: A A^T = I Substituting A^T = A into the above equation, we get: A A = I A^2 = I Thus, A^2 is equal to the identity matrix. Answer: Identity matrix

Question 12

For the following two (02) items: Let A = pmatrix & \ - & pmatrix . What is the value of the determinant of the matrix A^4?

Accepted Answer

The determinant of matrix A is given by: |A| = ( ) - (- ) |A| = ^2 + ^2 = 1 Using the property of determinants |A^n| = |A|^n, we have: |A^4| = |A|^4 = 1^4 = 1 Answer: 1

Question 13

What is the value of the determinant of the inverse of the matrix pmatrix -4 & -5 \ 2 & 2 pmatrix ?

Accepted Answer

Let A = pmatrix -4 & -5 \ 2 & 2 pmatrix . The determinant of matrix A is given by: |A| = (-4)(2) - (-5)(2) |A| = -8 + 10 = 2 The determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix, i.e., |A^ -1 | = 1 |A| . |A^ -1 | = 1 2 Answer: 1 2

Question 14

If f( ) = pmatrix & \ - & pmatrix , then what is \ f( )\ ^2 equal to?

Accepted Answer

Given f( ) = pmatrix & \ - & pmatrix Substituting = : f( ) = pmatrix & \ - & pmatrix = pmatrix -1 & 0 \ 0 & -1 pmatrix Squaring the matrix: \ f( )\ ^2 = pmatrix -1 & 0 \ 0 & -1 pmatrix pmatrix -1 & 0 \ 0 & -1 pmatrix = pmatrix 1 & 0 \ 0 & 1 pmatrix

Matrices — Previous Year Questions & Solutions