CBSE Class 12 Mathematics Matrices Assignment Set 02

This Class 12 CBSE Mathematics assignment on Matrices has 15 practice questions on types of matrices, matrix operations, transpose and symmetric and skew-symmetric matrices. Answers are given with every question for quick self-checking.

Practice Questions with Answers

Question 1. If \( \begin{bmatrix} x + y \\ x - y \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix} \times \begin{bmatrix} 1 \\ -2 \end{bmatrix} \), then find values of x and y.
Answer:
x = −1, y = 1

Question 2. If A, B are symmetric matrices of same order, then show that AB − BA is a skew-symmetric matrix.
Answer:
Since A′ = A and B′ = B, (AB − BA)′ = B′A′ − A′B′ = BA − AB = −(AB − BA). Hence AB − BA is skew-symmetric.

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

Question 4. If \( \begin{bmatrix} 2x + y & 4x \\ 5x - 7 & 4x \end{bmatrix} = \begin{bmatrix} 7 & 7y - 13 \\ y & x + 6 \end{bmatrix} \), then find the value of x and y.
Answer:
x = 2, y = 3

Question 5. 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 find \( A^3 \).
Answer:
\( \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \)

Question 6. If \( \begin{bmatrix} 2 & 1 & 3 \end{bmatrix}\begin{bmatrix} -1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} = A \), then write the order of matrix A.
Answer:
1 × 1

Question 7. If \( A = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix} \) such that \( A^2 = I \), then find the value of \( 1 - \alpha^2 - \beta\gamma \).
Answer:
0

Question 8. If \( A = \begin{bmatrix} 1 & 2 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & -2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \) and \( AB = I_3 \), then find the value of x + y.
Answer:
0

Question 9. If A and B are symmetric matrices, then show that BA − 2AB is neither symmetric nor skew-symmetric matrix.
Answer:
(BA − 2AB)′ = A′B′ − 2B′A′ = AB − 2BA, which in general is equal neither to BA − 2AB nor to −(BA − 2AB) = 2AB − BA. Hence BA − 2AB is neither symmetric nor skew-symmetric.

Question 10. Find a matrix which is both symmetric and skew-symmetric.
Answer:
Zero matrix

Question 11. Find the number of all possible matrices of order 3 × 3 with each entry 0 or 1.
Answer:
512

Question 12. If \( A' = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix} \) and \( B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix} \), then find (A + 2B)′.
Answer:
\( \begin{bmatrix} -4 & 5 \\ 1 & 6 \end{bmatrix} \)

Question 13. If matrix \( A = \begin{bmatrix} 3 & -3 \\ -3 & 3 \end{bmatrix} \) and \( A^2 = \lambda A \), then write the value of λ.
Answer:
λ = 6

Question 14. If \( A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \), then find \( A^2 - 5A + 8I \).
Answer:
\( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)

Question 15. If A is a matrix of order m × n and B is a matrix such that AB′ and B′A are both defined, then write the order of matrix B.
Answer:
m × n