## Matrices and Determinants Questions and Answers Part-6

1. $A=\begin{bmatrix}1 & 2 & 3 \\1 & 2 & 3 \\-1 & -2 & -3\end{bmatrix}$     then A is a nilpotent matrix of index
a) 2
b) 3
c) 4
d) 5

Explanation: A2 = O

2. If $A=\begin{bmatrix}\frac{1}{2}\left(e^{ix}+e^{-ix}\right) & \frac{1}{2}\left(e^{ix}-e^{-ix}\right) \\\frac{1}{2}\left(e^{ix}-e^{-ix}\right) & \frac{1}{2}\left(e^{ix}+e^{-ix}\right) \end{bmatrix}$       then $A^{-1}$ exists
a) for all real x
b) for positive real x only
c) for negative real x only
d) none of these

Explanation: |A| = 1 for each x

3. If $A=\begin{bmatrix}ab & b^{2} \\-a^{2} & -ab \end{bmatrix}$     then $A^{2}$ is equal
a) O
b) I
c) -I
d) None of the above

Explanation: Calculate directly

4.If A is $2 \times2$  matrix such that $A^{2}=O$  , then tr (A) is
a) 1
b) -1
c) 0
d) none of these

Explanation:

5. If $A=\begin{bmatrix}a & b \\c & d \end{bmatrix}$     such that A satisfies the relation $A^{2}-\left(a+d\right)A=O$     , then inverse of A is
a) I
b) A
c) (a+d) A
d) does not exist

Explanation: Use A2- (a + d)A + ad - bc = 0 to obtain ad - bc = 0

6. If $A=\begin{bmatrix}3 & 2 \\0 & 1 \end{bmatrix}$     , then $A^{-3}$ is
a) $\frac{1}{27}\begin{bmatrix}1 & -26 \\0 & -27 \end{bmatrix}$
b) $\frac{1}{27}\begin{bmatrix}-1 & -26 \\0 & -27 \end{bmatrix}$
c) $\frac{1}{27}\begin{bmatrix}1 & -26 \\0 & 27 \end{bmatrix}$
d) $\frac{1}{27}\begin{bmatrix}-1 & 26 \\0 & -27 \end{bmatrix}$

Explanation:

7. If A is a skew Hermitian matrix, then the main diagonal elements of A are all
a) zero
b) positive
c) negative
d) none of these

Explanation: None of these

8. If $A=\begin{bmatrix}1 & 2 & 1 \\0 & 1 & -1 \\3 & -1 & 1\end{bmatrix}$     then $A^{3}-3A^{2}-A-9I$     is equal to
a) O
b) I
c) A
d) $A^{2}$

Explanation: Calculate directly.

9.If $A=\begin{bmatrix}\omega & -\omega \\-\omega & \omega \end{bmatrix}$     and $B=\begin{bmatrix}1 & -1 \\-1 & 1 \end{bmatrix}$     ,then $A^{9}$ equals
a) 64 B
b) 32 B
c) 16 B
d) 256 B

Explanation: Use A = $\omega$ B.
10. If $A=\begin{bmatrix}2 & 3-i & -i \\3+i & -5 & 7+i \\i & 7-i & e\end{bmatrix}$      then A is