JEE Mathematics Matrices MCQs

Question. If \( A = \begin{bmatrix} 1 & -2 & 4 \\ 2 & 3 & 2 \\ 3 & 1 & 5 \end{bmatrix} \) and \( B = \begin{bmatrix} 0 & -2 & 4 \\ 1 & 3 & 2 \\ -1 & 1 & 5 \end{bmatrix} \) then \( A + B \) is
(a) \( \begin{bmatrix} 1 & -2 & 4 \\ 3 & 3 & 2 \\ 2 & 1 & 5 \end{bmatrix} \)
(b) \( \begin{bmatrix} 1 & -2 & 8 \\ 3 & 3 & 4 \\ 2 & 1 & 10 \end{bmatrix} \)
(c) \( \begin{bmatrix} 1 & -4 & 8 \\ 3 & 6 & 4 \\ 2 & 2 & 10 \end{bmatrix} \)
(d) None of the options
Answer: (c) \( \begin{bmatrix} 1 & -4 & 8 \\ 3 & 6 & 4 \\ 2 & 2 & 10 \end{bmatrix} \)

Question. If \( A^2 = 8A + kI \) where \( A = \begin{bmatrix} 1 & 0 \\ -1 & 7 \end{bmatrix} \) then \( k \) is
(a) 7
(b) -7
(c) 1
(d) -1
Answer: (b) -7

Question. The matrix \( \begin{bmatrix} \lambda & 7 & -2 \\ 4 & 1 & 3 \\ 2 & -1 & 2 \end{bmatrix} \) is a singular matrix if \( \lambda \) is
(a) \( \frac{2}{5} \)
(b) \( \frac{5}{2} \)
(c) -5
(d) None of the options
Answer: (a) \( \frac{2}{5} \)

Question. If the matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) then \( A^2 \) is
(a) \( \begin{bmatrix} a^2 & b^2 \\ c^2 & d^2 \end{bmatrix} \)
(b) \( \begin{bmatrix} a^2 + bc & ab + bd \\ ac + dc & bc + d^2 \end{bmatrix} \)
(c) nonexistent
(d) None of the options
Answer: (b) \( \begin{bmatrix} a^2 + bc & ab + bd \\ ac + dc & bc + d^2 \end{bmatrix} \)

Question. If \( A = \begin{bmatrix} \alpha & 0 \\ 1 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & 0 \\ 5 & 1 \end{bmatrix} \) such that \( A^2 = B \) then \( \alpha \) is
(a) 1
(b) -1
(c) 4
(d) None of the options
Answer: (d) None of the options

Question. If \( \begin{bmatrix} 2 & -3 \\ 1 & \lambda \end{bmatrix} \times \begin{bmatrix} 1 & 5 & \mu \\ 0 & 2 & -3 \end{bmatrix} = \begin{bmatrix} 2 & 4 & 1 \\ 1 & -1 & 13 \end{bmatrix} \) then
(a) \( \lambda = 3, \mu = 4 \)
(b) \( \lambda = 4, \mu = -3 \)
(c) no real values of \( \lambda, \mu \) are possible
(d) None of the options
Answer: (d) None of the options

Question. If \( AB = 0 \) where \( A = \begin{bmatrix} \cos^2 \theta & \cos \theta \sin \theta \\ \cos \theta \sin \theta & \sin^2 \theta \end{bmatrix} \) and \( B = \begin{bmatrix} \cos^2 \phi & \cos \phi \sin \phi \\ \cos \phi \sin \phi & \sin^2 \phi \end{bmatrix} \) then \( |\theta - \phi| \) is equal to
(a) 0
(b) \( \frac{\pi}{2} \)
(c) \( \frac{\pi}{4} \)
(d) \( \pi \)
Answer: (b) \( \frac{\pi}{2} \)

Question. If \( A = \begin{bmatrix} 0 & -4 & 1 \\ 2 & \lambda & -3 \\ 1 & 2 & -1 \end{bmatrix} \) then \( A^{-1} \) exists (i.e., A is invertible) if
(a) \( \lambda \neq 4 \)
(b) \( \lambda \neq 8 \)
(c) \( \lambda = 4 \)
(d) None of the options
Answer: (b) \( \lambda \neq 8 \)

Question. The reciprocal matrix of \( \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & -1 \\ 1 & 2 & 1 \end{bmatrix} \) is
(a) \( \begin{bmatrix} 3 & 4 & -2 \\ -1 & -1 & 1 \\ -1 & -2 & 1 \end{bmatrix} \)
(b) \( \begin{bmatrix} 3 & 4 & -2 \\ 1 & 1 & 1 \\ 1 & 2 & 1 \end{bmatrix} \)
(c) \( \begin{bmatrix} -3 & -1 & 1 \\ -4 & 1 & 2 \\ 2 & -1 & -1 \end{bmatrix} \)
(d) None of the options
Answer: (a) \( \begin{bmatrix} 3 & 4 & -2 \\ -1 & -1 & 1 \\ -1 & -2 & 1 \end{bmatrix} \)

Question. If \( A = \begin{bmatrix} 1 & -1 & 1 \\ 1 & 2 & 0 \\ 1 & 3 & 0 \end{bmatrix} \) then the value of \( |adj A| \) is equal to
(a) 5
(b) 0
(c) 1
(d) None of the options
Answer: (c) 1

Question. If \( A = \begin{bmatrix} \cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix} \) then \( A^{-1} \) is equal to
(a) \( A^T \)
(b) \( A \)
(c) \( adj A \)
(d) None of the options
Answer: (c) \( adj A \)

Question. If \( A = \begin{bmatrix} 4 & -1 & -4 \\ 3 & 0 & -4 \\ 3 & -1 & -3 \end{bmatrix} \) then \( A^2 \) is equal to
(a) \( A \)
(b) \( I \)
(c) \( A^T \)
(d) None of the options
Answer: (b) \( I \)

Question. If \( f(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix} \) then \( f(x + y) \) is equal to
(a) \( f(x) + f(y) \)
(b) \( f(x) - f(y) \)
(c) \( f(x) \cdot f(y) \)
(d) None of the options
Answer: (c) \( f(x) \cdot f(y) \)

Question. If \( A = \begin{bmatrix} 1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \end{bmatrix} \), \( B = \begin{bmatrix} \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \\ \omega & \omega^2 & 1 \end{bmatrix} \) and \( C = \begin{bmatrix} 1 \\ \omega \\ \omega^2 \end{bmatrix} \) where \( \omega \) is the complex cube root of 1 then \( (A + B)C \) is equal to
(a) \( \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \)
(b) \( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)
(c) \( \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \)
(d) \( \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \)
Answer: (a) \( \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \)

Question. If \( A = \begin{bmatrix} 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 \end{bmatrix} \) and \( B = \begin{bmatrix} a^2 & ab & ac \\ ba & b^2 & bc \\ ca & cb & c^2 \end{bmatrix} \) then \( AB \) is equal to
(a) 0
(b) \( I \)
(c) \( 2I \)
(d) None of the options
Answer: (a) 0

Question. If \( A \) be a matrix such that \( A \times \begin{bmatrix} 1 & -2 \\ 1 & 4 \end{bmatrix} = \begin{bmatrix} 6 & 0 \\ 0 & 6 \end{bmatrix} \) then \( A \) is
(a) \( \begin{bmatrix} 2 & 4 \\ 1 & -1 \end{bmatrix} \)
(b) \( \begin{bmatrix} -1 & 1 \\ 4 & 2 \end{bmatrix} \)
(c) \( \begin{bmatrix} 4 & 2 \\ -1 & 1 \end{bmatrix} \)
(d) None of the options
Answer: (c) \( \begin{bmatrix} 4 & 2 \\ -1 & 1 \end{bmatrix} \)

Question. The rank of the matrix \( \begin{bmatrix} -5 & 3 & 2 \\ 3 & 2 & -5 \\ 4 & -1 & -3 \end{bmatrix} \) is
(a) 3
(b) 2
(c) 1
(d) None of the options
Answer: (b) 2

Question. The rank of the matrix \( \begin{bmatrix} 1 & 2 & 3 \\ \lambda & 2 & 4 \\ 2 & -3 & 1 \end{bmatrix} \) is 3 if
(a) \( \lambda \neq \frac{18}{11} \)
(b) \( \lambda = \frac{18}{11} \)
(c) \( \lambda = -\frac{18}{11} \)
(d) None of the options
Answer: (a) \( \lambda \neq \frac{18}{11} \)

Question. The rank of the matrix \( \begin{bmatrix} 4 & 1 & 0 & 0 \\ 3 & 0 & 1 & 0 \\ 5 & 0 & 0 & 1 \end{bmatrix} \) is
(a) 4
(b) 3
(c) 2
(d) None of the options
Answer: (b) 3

Question. The system of equations \( x + y + z = 2 \), \( 2x - y + 3z = 5 \), \( x - 2y - z + 1 = 0 \) written in matrix form is
(a) \( \begin{bmatrix} x \\ y \\ z \end{bmatrix} \begin{bmatrix} 1 & 1 & 1 \\ 2 & -1 & 3 \\ 1 & -2 & -1 \end{bmatrix} = \begin{bmatrix} 2 \\ 5 \\ -1 \end{bmatrix} \)
(b) \( \begin{bmatrix} 1 & 1 & 1 \\ 2 & -1 & 3 \\ 1 & -2 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -2 \\ -5 \\ 1 \end{bmatrix} \)
(c) \( \begin{bmatrix} 1 & 1 & 1 \\ 2 & -1 & 3 \\ 1 & -2 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 2 \\ 5 \\ -1 \end{bmatrix} \)
(d) None of the options
Answer: (c) \( \begin{bmatrix} 1 & 1 & 1 \\ 2 & -1 & 3 \\ 1 & -2 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 2 \\ 5 \\ -1 \end{bmatrix} \)

Question. If \( \begin{bmatrix} 1 & x & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ x \end{bmatrix} = 0 \) then \( x \) is
(a) 2
(b) -2
(c) 14
(d) None of the options
Answer: (b) -2

Question. If \( \begin{bmatrix} x + y & y \\ 2x & x - y \end{bmatrix} \begin{bmatrix} 2 \\ -1 \end{bmatrix} = \begin{bmatrix} 3 \\ 2 \end{bmatrix} \) then \( x.y \) is equal to
(a) -5
(b) 5
(c) 4
(d) 6
Answer: (a) -5

Choose the correct options. One or more options may be correct.

Question. \( \begin{bmatrix} 1 & -2 & 3 \\ 2 & -1 & 4 \\ 3 & 4 & 1 \end{bmatrix} \) is a
(a) rectangular matrix
(b) singular matrix
(c) square matrix
(d) nonsingular matrix
Answer: (c) square matrix (d) nonsingular matrix

Question. If \( A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \\ 0 & 6 \end{bmatrix} \) and \( B = \begin{bmatrix} 5 & 4 & 6 \\ 4 & 1 & 2 \\ -5 & -1 & 1 \end{bmatrix} \) then
(a) \( A + B \) exists
(b) \( AB \) exists
(c) \( BA \) exists
(d) None of the options
Answer: (c) BA exists

Question. If \( A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix} \) then
(a) \( A^3 = 9A \)
(b) \( A^3 = 27A \)
(c) \( A^2 = 3A \)
(d) \( A^{-1} \) does not exist
Answer: (a) \( A^3 = 9A \) (c) \( A^2 = 3A \) (d) \( A^{-1} \) does not exist