## Matrices and Determinants Questions and Answers Part-7

1. If A and B are two square matrices of the same order and m is a positive integer, then
$\left(A+B\right)^{m}=^{m}C_{0}A^{m}+^{m}C_{1}A^{m-1}B+^{m}C_{2}A^{m-2}B^{2}+....+^{m}C_{m-1}AB^{m-1}+^{m}C_{m}B^{m}$
if
a) AB = BA
b) AB + BA = O
c) $A^{m}=O, B^{m}=O$
d) none of these

Explanation: Binomial theorem is applicable if and only if AB = BA.

2. If $A\left(\theta\right)=\begin{bmatrix}\cos\theta & -\sin\theta & 0 \\\sin\theta & \cos\theta & 0 \\0 & 0 & 0\end{bmatrix}$         then $A\left(\theta\right)^{3}$ will be a null matrix if and only if
a) $\theta=\left(2k+1\right)\pi/3,k \epsilon I$
b) $\theta=\left(4k-1\right)\pi/3,k \epsilon I$
c) $\theta=\left(3k-1\right)\pi/4,k \epsilon I$
d) none of these

Explanation: Use A( $\theta$ )3 = A(3 $\theta$ )

3. If A and B are two non-singular matrices such that AB = C, then |B| is equal to
a) $\frac{\mid C\mid}{\mid A\mid }$
b) $\frac{\mid A\mid}{\mid C\mid }$
c) $\mid C\mid$
d) none of these

Explanation: Use |AB| = |A| |B|

4. If the system of equations ax + y= 3, x + 2y = 3, 3x + 4y = 7 is consistent, then value of a is given by
a) 2
b) 1
c) -1
d) 0

Explanation: From the last two equations we get x = 1, y = 1. This gives a = 2.

5. If the system of equations x + 2y – 3z = 1, (p + 2)z = 3, (2p + 1) y + z = 2 is consistent, then the value of p is
a) -2
b) -1/2
c) 0
d) 2

Explanation:

6. The system of linear equations x + y + z = 2, 2x + y – z = 3, 3x + 2y + kz = 4 has a unique solution if
a) $k\neq 0$
b) – 1 < k < 1
c) – 2 < k < 2
d) k = 0

Explanation:

7. If A, B and A + B are non-singular matrices, then $\left(A^{-1}+B^{-1}\right)\left[A-A\left(A+B\right)^{-1}A\right]$
a) O
b) I
c) A
d) B

Explanation:

8. If $\omega \neq 1$  is cube root of unity, and $A=\begin{bmatrix}1 & \omega & \omega^{2} \\\omega & \omega^{2} & 1 \\\omega^{2} & 1 & \omega\end{bmatrix}$         is
a) symmetric
b) skew symmetric
c) singular
d) Both a and c

Explanation: |A| = 0

9. Suppose A and B are two $3 \times3$  non-singular matrices such that $\left(AB\right)^{k}=A^{k}B^{k}$
for k = 2017, 2018, 2019, then
a) $AB^{-1}A^{-1}=B^{-1}$
b) $A^{-2}BA^{2}=\left(A^{-1}BA\right)^{2}$
c) AB = BA
d) All of the above

Explanation: $AB^{-1}A^{-1}=B^{-1}$
10. If A and B are $3 \times3$  matrices and $\mid A\mid\neq 0$   , then
a) $\mid AB\mid= 0\Rightarrow \mid B\mid=0$
b) $\mid AB\mid\neq 0\Rightarrow \mid B\mid\neq0$
c) $\mid A^{-1}\mid= \mid A\mid^{-1}$
Explanation: $\mid A^{-1}\mid= \mid A\mid^{-1}$