CBSE Class 12 Mathematics Matrices and Determinants MCQs Set 04

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: (c) \( A^2 - B^2 + BA - AB \)

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

Question. The matrix \( \begin{bmatrix} 0 & -5 & 8 \\ 5 & 0 & 12 \\ -8 & -12 & 0 \end{bmatrix} \) is a
(a) diagonal matrix
(b) symmetric matrix
(c) skew symmetric matrix
(d) scalar matrix
Answer: (c) skew symmetric matrix

Question. If \( A \) is matrix of order \( m \times n \) and \( B \) is a matrix such that \( AB' \) and \( B'A \) are both defined, the order of matrix \( B \) is
(a) \( m \times m \)
(b) \( n \times n \)
(c) \( n \times m \)
(d) \( m \times n \)
Answer: (d) \( m \times n \)

Question. If \( A \) and \( B \) are matrices of same order, then \( (AB' - BA') \) is a
(a) skew-symmetric matrix
(b) null matrix
(c) symmetric matrix
(d) unit matrix
Answer: (a) skew-symmetric matrix

Question. If \( A \) is square matrix such that \( A^2 = I \), then \( (A - I)^3 + (A + I)^3 - 7A \) is equal to
(a) \( A \)
(b) \( I - A \)
(c) \( I + A \)
(d) \( 3A \)
Answer: (a) \( A \)

Question. For any two matrices \( A \) and \( B \), we have
(a) \( AB = BA \)
(b) \( AB \neq BA \)
(c) \( AB = O \)
(d) None of the options
Answer: (d) None of the options

Question. The number of all possible matrices of order \( 3 \times 3 \) with each entry 0 or 1 is
(a) 27
(b) 18
(c) 81
(d) 512
Answer: (d) 512

Question. The restriction on \( n \), \( k \) and \( p \) so that \( PY + WY \) will be defined are
(a) \( k = 3 \), \( p = n \)
(b) \( k \) is arbitrary, \( p = 2 \)
(c) \( p \) is arbitrary, \( k = 3 \)
(d) \( k = 2 \), \( p = 3 \)
Answer: (a) \( k = 3 \), \( p = n \)

Question. If \( n = p \), then the order of the matrix \( 7X - 5Z \) is
(a) \( p \times 2 \)
(b) \( 2 \times n \)
(c) \( n \times 3 \)
(d) \( p \times n \)
Answer: (b) \( 2 \times n \)

Question. If \( \begin{bmatrix} x - y & 2 \\ x & 5 \end{bmatrix} = \begin{bmatrix} 2 & 2 \\ 3 & 5 \end{bmatrix} \), then value of \( y \) is
(a) 1
(b) 3
(c) 2
(d) 5
Answer: (a) 1

Question. If \( A = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix} \) is such that \( A^2 = I \), then
(a) \( 1 + \alpha^2 + \beta\gamma = 0 \)
(b) \( 1 - \alpha^2 + \beta\gamma = 0 \)
(c) \( 1 - \alpha^2 - \beta\gamma = 0 \)
(d) \( 1 + \alpha^2 - \beta\gamma = 0 \)
Answer: (c) \( 1 - \alpha^2 - \beta\gamma = 0 \)

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 \( A = \begin{bmatrix} 0 & 0 & 5 \\ 0 & 5 & 0 \\ 5 & 0 & 0 \end{bmatrix} \)
(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. The matrix \( P = \begin{bmatrix} 0 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 0 \end{bmatrix} \) is a
(a) square matrix
(b) diagonal matrix
(c) unit matrix
(d) none
Answer: (a) square matrix

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

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. On using elementary column operations \( C_2 \rightarrow C_2 - 2C_1 \) in the following matrix equation \( \begin{bmatrix} 1 & -3 \\ 2 & 4 \end{bmatrix} = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} \), we have:
(a) \( \begin{bmatrix} 1 & -5 \\ 0 & 4 \end{bmatrix} = \begin{bmatrix} 1 & -1 \\ -2 & 2 \end{bmatrix} \begin{bmatrix} 3 & -5 \\ 2 & 0 \end{bmatrix} \)
(b) \( \begin{bmatrix} 1 & -5 \\ 0 & 4 \end{bmatrix} = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 3 & -5 \\ 0 & 2 \end{bmatrix} \)
(c) \( \begin{bmatrix} 1 & -5 \\ 2 & 0 \end{bmatrix} = \begin{bmatrix} 1 & -3 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 3 & 1 \\ -2 & 4 \end{bmatrix} \)
(d) \( \begin{bmatrix} 1 & -5 \\ 2 & 0 \end{bmatrix} = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 3 & -5 \\ 2 & 0 \end{bmatrix} \)
Answer: (d) \( \begin{bmatrix} 1 & -5 \\ 2 & 0 \end{bmatrix} = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 3 & -5 \\ 2 & 0 \end{bmatrix} \)

Question. If the matrix \( AB \) is zero, then
(a) It is not necessary that either \( A = O \) or \( B = O \)
(b) \( A = O \) or \( B = O \)
(c) \( A = O \) and \( B = O \)
(d) All the statements are wrong
Answer: (a) It is not necessary that either \( A = O \) or \( B = O \)

Assertion-Reason Questions [1 mark]

The following questions consist of two statements—Assertion(A) and Reason(R). Answer these questions selecting the appropriate option given below:
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.

Question. Assertion (A) : A matrix \( A = \begin{bmatrix} 1 & 2 & 0 & 3 \end{bmatrix} \) is a row matrix of order \( 1 \times 4 \).
Reason (R) : A matrix having one row and any number of column is called a row matrix.
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.
Solution:
We have, \( A = \begin{bmatrix} 1 & 2 & 0 & 3 \end{bmatrix} \) matrix which has one row and four columns.
Therefore, it is a row matrix.
Clearly, both Assertion (A) and Reason (R) are True and Reason (R) is the correct explanation of Assertion (A).
Hence, (a) is the correct option.

Question. Assertion (A) : If \( \begin{bmatrix} x^2 - 4x & x^2 \\ x^2 & x^3 \end{bmatrix} = \begin{bmatrix} -3 & 1 \\ -x + 2 & 1 \end{bmatrix} \), then the value of \( x = 1 \).
Reason (R) : Two matrices \( A = [a_{ij}]_{m \times n} \) and \( B = [b_{ij}]_{m \times n} \) of same order \( m \times n \) are equal, if \( a_{ij} = b_{ij} \) for all \( i = 1, 2, 3, ....m \) and \( j = 1, 2, 3, ... n \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.
Solution:
We have, \( x^2 - 4x = -3 \)
\( \implies \) \( x^2 - 4x + 3 = 0 \)
\( \implies \) \( x^2 - 3x - x + 3 = 0 \)
\( \implies \) \( x(x - 3) - 1(x - 3) = 0 \)
\( \implies \) \( (x - 1)(x - 3) = 0 \)
\( \implies \) \( x = 1, 3 \)
and, \( x^2 = 1 \)
\( \implies \) \( x = \pm 1 \)
Also, \( x^2 = -x + 2 \)
\( \implies \) \( x^2 + x - 2 = 0 \)
\( \implies \) \( x^2 + 2x - x - 2 = 0 \)
\( \implies \) \( x(x + 2) - 1(x + 2) = 0 \)
\( \implies \) \( (x + 2)(x - 1) = 0 \)
\( \implies \) \( x = 1, -2 \)
and, \( x^3 = 1 \)
\( \implies \) \( x^3 - 1 = 0 \)
\( \implies \) \( (x - 1)(x^2 + x + 1) = 0 \)
\( \implies \) \( x = 1, x^2 + x + 1 = 0 \)
\( \implies \) \( x = \frac{-1 \pm \sqrt{-3}}{2} \)
\( \implies \) \( x = \frac{-1 \pm \sqrt{3}i}{2} = \omega, \omega^2 \)
\( \therefore x = 1, \omega, \omega^2 \)
So, common value of \( x = 1 \).
Clearly, both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Hence, (a) is the correct option.

Question. Assertion (A) : If \( A \) and \( B \) are symmetric matrices of same order then \( AB - BA \) is also a symmetric matrix.
Reason (R) : Any square matrix \( A \) is said to be skew-symmetric matrix if \( A = -A^T \), where \( A^T \) is the transpose of matrix \( A \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (d) A is false but R is true.
Solution:
We have, \( AB - BA = P \quad (\text{Let}) \)
\( \therefore \quad P^T = (AB - BA)^T = (AB)^T - (BA)^T \)
\( = B^T A^T - A^T B^T = -(A^T B^T - B^T A^T) \)
\( \implies \) \( P^T = -(AB - BA) = -P \)
\( \implies \) \( P = -P^T \)
\( \therefore \quad P \text{ is skew-symmetric matrix.} \)
\( \implies \) \( AB - BA \text{ is skew-symmetric matrix.} \)
Clearly, Assertion (A) is false and Reason (R) is true.
Hence, (d) is the correct option.

Question. Assertion (A) : If \( A \) is a square matrix of order \( 3 \times 3 \), and 2 is any scalar then the value of \( |2A| = 8|A| \).
Reason (R) : If \( k \) is a scalar and \( A \) is a square matrix of order \( n \times n \). Then \( |kA| = k^{n-1} |A| \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (c) A is true but R is false.
Solution:
As we know that if \( k \) is any scalar and \( A \) be any square matrix of order \( n \times n \) than
\( |kA| = k^n \cdot |A| \)
\( \therefore \quad |2A| = 2^3 \cdot |A| \quad (\because A \text{ is of order } 3 \times 3 \text{ in Assertion (A)}) \)
\( = 8|A| \)
Clearly, Assertion (A) is true and Reason (R) is false.
Hence, (c) is the correct option.

Question. Assertion (A) : If \( A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} \), than \( A^2 = \begin{bmatrix} \cos 2\theta & \sin 2\theta \\ -\sin 2\theta & \cos 2\theta \end{bmatrix} \).
Reason (R) : If \( A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} \), than \( A^n = \begin{bmatrix} \cos n\theta & \sin n\theta \\ -\sin n\theta & \cos n\theta \end{bmatrix} \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.
Solution:
Clearly, both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Hence, (a) is the correct option.