CBSE Class 12 Mathematics Matrices and Determinants MCQs Set 05

Question. If A and B are square matrices of the same order, then (A + B) (A – B) is equal to:
(a) \( A^2 - B^2 \)
(b) \( A^2 - BA - AB - B^2 \)
(c) \( A^2 - B^2 + BA - AB \)
(d) \( A^2 - BA + B^2 + AB \)
Answer: (b) \( A^2 - BA - AB - B^2 \)

Question. If \( A = \begin{bmatrix} 2 & -1 & 3 \\ -4 & 5 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} 2 & 3 \\ 4 & -2 \\ 1 & 5 \end{bmatrix} \), then
(a) only AB is defined
(b) only BA is defined
(c) AB and BA both are defined
(d) AB and BA both are not defined
Answer: (c) AB and BA both are defined

Question. The matrix \( A = \begin{bmatrix} 0 & 0 & 5 \\ 0 & 5 & 0 \\ 5 & 0 & 0 \end{bmatrix} \) is a:
(a) scalar matrix
(b) diagonal matrix
(c) unit matrix
(d) square matrix
Answer: (d) square matrix

Question. If A and B are symmetric matrices of the same order, then (AB′ – BA′) is a
(a) Skew symmetric matrix
(b) Null matrix
(c) Symmetric matrix
(d) None of the options
Answer: (a) Skew symmetric matrix

Question. If A = [2 – 3 4], \( B = \begin{bmatrix} 3 \\ 2 \\ 2 \end{bmatrix} \), X = [1 2 3] and \( Y = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} \), then AB + XY equals :
(a) [28]
(b) [24]
(c) 28
(d) 24
Answer: (a) [28]

Question. If \( [x - 1] \begin{bmatrix} 1 & 0 \\ -2 & 0 \end{bmatrix} = 0 \), then x equals: 
(a) 0
(b) – 2
(c) – 1
(d) 2
Answer: (b) – 2

Question. Which of the following is a \( I_2 \) matrix?
(a) \( \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \)
(b) \( \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \)
(c) \( \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \)
(d) \( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)
Answer: (d) \( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)

Question. Two matrices A and B are multiplied to get AB if,
(a) both are rectangular
(b) both have same order
(c) no. of columns of A is equal to no. of rows of B
(d) no. of rows of A is equal to no. of columns of B
Answer: (c) no. of columns of A is equal to no. of rows of B

Question. If A is a matrix of order 3 × 4, then each row of A has:
(a) 3 elements
(b) 12 elements
(c) 7 elements
(d) 4 elements
Answer: (d) 4 elements

Question. The matrix \( A = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix} \) is a:
(a) Scalar matrix
(b) Diagonal matrix
(c) Unit matrix
(d) Square matrix
Answer: (a) Scalar matrix

Question. If A is a square matrix of order 3, such that A (adj A) = 10 I, then |adj A| is equal to:
(a) 1
(b) 10
(c) 100
(d) 101
Answer: (c) 100

Question. The number of all possible matrices of order 2 × 3 with each entry 0 or 1 is:
(a) 64
(b) 12
(c) 36
(d) None of the options
Answer: (a) 64

Question. The matrix \( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix} \) is a :
(a) Identity matrix
(b) Scalar matrix
(c) Skew-symmetric matrix
(d) Diagonal matrix
Answer: (d) Diagonal matrix

Question. For any 2 × 2 matrix, if \( A(\text{adj. } A) = \begin{bmatrix} 10 & 0 \\ 0 & 10 \end{bmatrix} \), then |A| is equal to :
(a) 20
(b) 100
(c) 10
(d) 0
Answer: (c) 10

Question. A is a scalar matrix with scalar k ≠ 0 of order 3. Then \( A^{-1} \) is:
(a) \( \begin{bmatrix} 1/k & 0 & 0 \\ 0 & 1/k & 0 \\ 0 & 0 & 1/k \end{bmatrix} \)
(b) \( \begin{bmatrix} 0 & 0 & k \\ 0 & k & 0 \\ k & 0 & 0 \end{bmatrix} \)
(c) \( \begin{bmatrix} 1/k^2 & 0 & 0 \\ 0 & 1/k^2 & 0 \\ 0 & 0 & 1/k^2 \end{bmatrix} \)
(d) \( \begin{bmatrix} k & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & k \end{bmatrix} \)
Answer: (a) \( \begin{bmatrix} 1/k & 0 & 0 \\ 0 & 1/k & 0 \\ 0 & 0 & 1/k \end{bmatrix} \)

Question. If A is a 3 × 2 matrix, B is a 3 × 3 matrix and C is a 2 × 3 matrix, then the elements in A, B and C are respectively:
(a) 6, 9, 8
(b) 6, 9, 6
(c) 9, 6, 6
(d) 6, 6, 9
Answer: (b) 6, 9, 6

Question. If a matrix has 8 elements, then which of the following will not be a possible order of the matrix?
(a) 1 × 8
(b) 2 × 4
(c) 4 × 2
(d) 4 × 4
Answer: (d) 4 × 4

Question. Total number of possible matrices of order 3 × 3 with each entry 2 or 0 is: 
(a) 9
(b) 27
(c) 81
(d) 512
Answer: (d) 512

Question. The matrix \( P = \begin{bmatrix} 0 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 0 \end{bmatrix} \) is not a: 
(a) square matrix
(b) diagonal matrix
(c) unit matrix
(d) None of the options
Answer: (c) unit matrix

Question. Which of the given values of x and y make the following pair of matrices equal: \( \begin{bmatrix} 3x+7 & 5 \\ y+1 & 2-3x \end{bmatrix} , \begin{bmatrix} 0 & y-2 \\ 8 & 4 \end{bmatrix} \) 
(a) x = –1/3, y = 7
(b) not possible to find
(c) y = 7, x = –2/3
(d) x = –1/3, y = –2/3
Answer: (b) not possible to find

Question. If \( A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \), \( B = \begin{bmatrix} 1 & 3 & 2 \\ 4 & 3 & 1 \end{bmatrix} \), \( C = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \) and \( D = \begin{bmatrix} 4 & 6 & 8 \\ 5 & 7 & 9 \end{bmatrix} \), then which of the following is defined? 
(a) A + B
(b) B + C
(c) C + D
(d) B + D
Answer: (d) B + D

Question. If \( \begin{bmatrix} 1 & 2 \\ -2 & -b \end{bmatrix} + \begin{bmatrix} a & 4 \\ 3 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 1 & 0 \end{bmatrix} \), then \( a^2 + b^2 \) is equal to:
(a) 20
(b) 22
(c) 12
(d) 10
Answer: (a) 20

Question. If \( A = \begin{bmatrix} 0 & 2 \\ 3 & -4 \end{bmatrix} \) and \( kA = \begin{bmatrix} 0 & 3a \\ 2b & 24 \end{bmatrix} \) then the values of k, a, b are respectively:
(a) – 6, – 12, – 18
(b) – 6, 4, 9
(c) – 6, – 4, – 9
(d) – 6, 12, 18
Answer: (c) – 6, – 4, – 9

Question. The product \( \begin{bmatrix} a & b \\ -b & a \end{bmatrix} \begin{bmatrix} a & -b \\ b & a \end{bmatrix} \) is equal to:
(a) \( \begin{bmatrix} a^2+b^2 & 0 \\ 0 & a^2+b^2 \end{bmatrix} \)
(b) \( \begin{bmatrix} (a+b)^2 & 0 \\ 0 & (a+b)^2 \end{bmatrix} \)
(c) \( \begin{bmatrix} a^2+b^2 & 0 \\ a^2+b^2 & 0 \end{bmatrix} \)
(d) \( \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \)
Answer: (a) \( \begin{bmatrix} a^2+b^2 & 0 \\ 0 & a^2+b^2 \end{bmatrix} \)

Question. If the product of two matrices is a zero matrix, then :
(a) atleast one of the matrix is a zero matrix
(b) both the matrices are zero matrices
(c) it is not necessary that one of the matrices is a zero matrix
(d) None of the options
Answer: (c) it is not necessary that one of the matrices is a zero matrix

Question. If \( A = \begin{bmatrix} 2 & -1 & 3 \\ -4 & 5 & 1 \end{bmatrix}_{2 \times 3} \) and \( B = \begin{bmatrix} 2 & 3 \\ 4 & -2 \\ 1 & 5 \end{bmatrix}_{3 \times 2} \), then :
(a) only AB is defined
(b) only BA is defined
(c) AB and BA both are defined
(d) AB and BA both are not defined
Answer: (c) AB and BA both are defined

Question. The set of all 2 × 2 matrices which is commutative with the matrix \( \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} \) with respect to matrix multiplication is:
(a) \( \begin{bmatrix} p & q \\ r & r \end{bmatrix} \)
(b) \( \begin{bmatrix} p & q \\ q & r \end{bmatrix} \)
(c) \( \begin{bmatrix} p-q & q \\ q & r \end{bmatrix} \)
(d) \( \begin{bmatrix} p & q \\ q & p-q \end{bmatrix} \)
Answer: (d) \( \begin{bmatrix} p & q \\ q & p-q \end{bmatrix} \)

Question. If A is matrix of order m × n and B is matrix such that AB’ and B’ A are both defined, then order of matrix B is: [NCERT Exemplar]
(a) m × n
(b) n × n
(c) n × m
(d) m × n
Answer: (d) m × n

Question. If A and B are square matrices of the same order and AB = 3I, then \( A^{-1} \) is equal to:
(a) 3B
(b) \( \frac{1}{3} B \)
(c) \( 3B^{-1} \)
(d) \( \frac{1}{3} B^{-1} \)
Answer: (b) \( \frac{1}{3} B \)

Question. If \( A = \begin{bmatrix} \sin^{-1}(x\pi) & \tan^{-1}(x/\pi) \\ \sin^{-1}(x/\pi) & \cot^{-1}(\pi x) \end{bmatrix} \) and \( B = \frac{1}{\pi} \begin{bmatrix} -\cos^{-1}(x\pi) & \tan^{-1}(x/\pi) \\ \sin^{-1}(x/\pi) & -\tan^{-1}(\pi x) \end{bmatrix} \), then A – B is equal to: 
(a) I
(b) O
(c) 2I
(d) \( \frac{1}{2} I \)
Answer: (d) \( \frac{1}{2} I \)

Question. If A and B are two matrices of the order 3 × m and 3 × n, respectively, and m = n, then the order of matrix (5A – 2B) is: 
(a) m × 3
(b) 3 × 3
(c) m × n
(d) 3 × n
Answer: (d) 3 × n

Question. If \( A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \), then \( A^2 \) is equal to: 
(a) \( \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \)
(b) \( \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix} \)
(c) \( \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix} \)
(d) \( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)
Answer: (d) \( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)

Question. If matrix \( A = [a_{ij}]_{2 \times 2} \) where \( a_{ij} = \begin{cases} 1 & \text{if } i \neq j \\ 0 & \text{if } i = j \end{cases} \), then \( A^2 \) is equal to: 
(a) I
(b) A
(c) 0
(d) None of the options
Answer: (a) I

Question. The matrix \( \begin{bmatrix} 1 & 2 & 4 \\ 2 & 5 & 6 \\ 4 & 6 & 7 \end{bmatrix} \) is a 
(a) identity matrix
(b) symmetric matrix
(c) skew symmetric matrix
(d) none of the options
Answer: (b) symmetric matrix

Question. \( A = [a_{ij}]_{m \times n} \) is a square matrix, if
(a) m < n
(b) m > n
(c) m = n
(d) None of the options
Answer: (c) m = n