Matrices and Determinants Questions and Answers Part-8

1. If A and B are two symmetric matrices of the same order, then
a) A + B is symmetric
b) A - B is symmetric
c) BAB is symmetric
d) All of the above

Explanation: A - B is symmetric

2. If A and B are two matrices of the same order, then
a) $A+A'$  is symmetric
b) $A^{2}=I\Leftrightarrow \left(A+I\right)\left(A-I\right)=O$
c) $\left(A'\right)'=A$
d) All of the above

Explanation: $\left(A'\right)'=A$

3. If $A=\left(a_{ij}\right)_{3\times 3}$   is a skew symmetric matrix, then
a) $a_{ii}=0\forall i$
b) A + A' is a null matrix
c) |A| = 0
d) All of the above

Explanation: |A| = 0

4. If A, B and C are three square matrices of the same order, then $AB=AC\Rightarrow B=C$     if
a) $\mid A\mid \neq 0$
b) A is invertible
c) A is orthogonal
d) All of the above

Explanation: |A| $\neq$ 0
A-1 exists

5. If $\alpha,\beta,\gamma$  are three real numbers and $A=\begin{bmatrix}1 & \cos\left(\alpha-\beta\right) & \cos\left(\alpha-\gamma\right) \\\cos\left(\beta-\alpha\right) & 1 & \cos\left(\beta-\gamma\right) \\\cos\left(\gamma-\alpha\right) & \cos\left(\gamma-\beta\right) & 1\end{bmatrix}$
then
a) A is symmetric
b) A is not invertible
c) A is singular
d) All of the above

Explanation:

6. Let $C_{k}=^{n}C_{k}$   for $0\leq k \leq n$   and
$A_{k}=\begin{bmatrix}C_{k-1}^{2 }& 0 \\0 & C_{k}^{2 } \end{bmatrix}$     for $k\geq 1$  and $A_{1}+A_{2}+....+A_{n}=\begin{bmatrix}k_{1} & 0 \\0 & k_{2} \end{bmatrix}$
then
a) $k_{1}=k_{2}$
b) $k_{1}+k_{2}=^{2n}C_{2n}+1$
c) $k_{1}=^{2n}C_{n}-1$
d) Both a and c

Explanation:

7. If $A\left(\theta\right)=\begin{bmatrix}\sin\theta & i\cos\theta \\i \cos\theta & \sin\theta \end{bmatrix}$     then
a) $A\left(\theta\right)$  is invertible for all $\theta\epsilon R$
b) $A\left(\theta\right)+A\left(\pi+\theta\right)$    is a null matrix
c) $A\left(\theta\right)^{-1}=A\left(\pi-\theta\right)$
d) All of the above

Explanation:

8. D is a $3 \times3$  diagonal matrix. Which of the following statements is not true?
a) D' = D
b) $D^{2}$ is a diagonal matrix
c) $D^{-1}$ if exists is a diagonal matrix
d) All of the above

Explanation: AD = DA for each $3 \times3$  matrix if and only if D is a scalar matrix, and D-1 if exists is a diagonal matrix

9. If A is an invertible matrix, then which of the followings are true
a) $A \neq O$
b) adj.$A \neq O$
c) $\mid A\mid \neq O$
d) All of the above

Explanation: adj.$A \neq O$
10. If $A=\begin{bmatrix}1 & -1 & 1 \\2 & -1 & 0 \\1 & 0 & 0\end{bmatrix}$     then
a) $A^{3}=I$
b) $A^{-1}=A^{2}$
c) $A^{n}=A\forall n\neq 4$