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) \[A^{n}=A\forall n\neq 4\]
d) Both a and b
Explanation:
