## Matrices and Determinants Questions and Answers Part-2

1. The inverse of a symmetric matrix (if it exists) is
a) a symmetric matrix
b) a skew symmetric matrix
c) a diagonal matrix
d) none of these

Answer: a
Explanation: Let A be an invertible symmetric matrix

2. The inverse of a skew symmetric matrix (if it exists) is
a) a symmetric matrix
b) a skew symmetric matrix
c) a diagonal matrix
d) none of these

Answer: b
Explanation: We have A' = – A

3. The inverse of a skew symmetric matrix of odd order is
a) a symmetric matrix
b) a skew symmetric matrix
c) diagonal matrix
d) does not exist

Answer: d
Explanation: Let A be a skew symmetric, matrix of order n

4. If A is an orthogonal matrix, then |A| is
a) 1
b) -1
c) $\pm1$
d) 0

Answer: c
Explanation: As A is an orthogonal matrix

5. If A is a $3\times3$  non-singular matrix, then adj (adj A) is equal to
a) $\mid A\mid A$
b) $\mid A\mid^{2} A$
c) $\mid A\mid^{-1} A$
d) 0

Answer: a
Explanation:

6. If A and B are two square matrices such that $B=-A^{-1} BA$   , then $\left(A+B \right)^{2}$   is equal to
a) O
b) $A^{2}+B^{2}$
c) $A^{2}+2AB+B^{2}$
d) A+B

Answer: b
Explanation: As B = – A–1 BA, we get AB = – BA

7. 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: b
Explanation:

8. The value of x for which the matrix $A=\begin{bmatrix}2 & 0 & 7 \\0 & 1 & 0 \\1 & -2 & 1\end{bmatrix}$     is inverse of $B=\begin{bmatrix}-x & 14x & 7x \\0 & 1 & 0 \\x & -4x & -2x\end{bmatrix}$
is
a) $\frac{1}{2}$
b) $\frac{1}{3}$
c) $\frac{1}{4}$
d) $\frac{1}{5}$

Answer: d
Explanation:

9. If $A\left(\alpha,\beta\right)=\begin{bmatrix}\cos\alpha & \sin\alpha & 0 \\-\sin\alpha & \cos\alpha & 0 \\0 & 0 & e^{\beta}\end{bmatrix}$     , then $A\left(\alpha,\beta\right)^{-1}$   is equal to
a) $A\left(-\alpha,\beta\right)$
b) $A\left(-\alpha,-\beta\right)$
c) $A\left(\alpha,-\beta\right)$
d) $A\left(\alpha,\beta\right)$

Answer: b
Explanation:

10. If A is a $3\times 3$  skew-symmetric matrix, then trace of A is equal to
a) 1
b) 3
c) -1
d) $\mid A\mid$

Answer: d
Explanation: As A is a skew symmetric matrix