## Matrices and Determinants Questions and Answers Part-5

1.If A is matrix of size $n \times n$  such that $A^{2}+A+2I=O$   , then
a) A is non-singular
b) $A\neq O$
c) $\mid A\mid\neq 0$
d) All of the above

Explanation: We have A(A + I) = – 2I

2. Let $\alpha=\pi/5$   and $A=\begin{bmatrix}\cos\alpha & \sin\alpha \\-\sin\alpha & \cos\alpha \end{bmatrix}$     then $B=A+A^{2}+A^{3}+A^{4}$
is
a) singular
b) non-singular
c) skew-symmetric
d) Both b and c

Explanation:

3. Let A, B be two $2 \times2$  matrices.
Let $\alpha= det\left(A\right) + det\left(B\right) – det\left(A + B\right)$
and $\beta= tr\left(AB\right) -\left(trA\right)\left(trB\right)$      , then
a) $\alpha= \beta$
b) $\alpha+ \beta=0$
c) $\alpha^{2}= \beta$
d) $\alpha= \beta^{2}$

Explanation:

4. Let A be a $3 \times3$  matrix with real entries such that sum of the entries in each column of A is 1, then sum of the entries of $A^{2018}$  is
a) 1
b) 3
c) 2018
d) 6054

Explanation:

5. If $A=\begin{bmatrix}0 & -1 & 2 \\1 & 0 & 3 \\-2 & -3 & 0\end{bmatrix}$        then A + 2A' equals
a) A
b) A'
c) -A'
d) 2A

Explanation: Use A = - A'

6. Let $A_{t}=\begin{bmatrix}1 & 3 & 2 \\2 & 5 & t \\4 & 7-t & -6\end{bmatrix}$
then the value(s) of t for which inverse of $A_{t}$ does not exist.
a) -2,1
b) 3,2
c) 2,-3
d) 3,-1

Explanation: Set |At| = 0

7. Let a, b, $c\epsilon R$  be such that a + b + c > 0 and abc = 2. Let
$A=\begin{bmatrix}a & b & c \\b & c & a \\c & a & b\end{bmatrix}$
If $A^{2}=I$  , then value of $a^{3}+b^{3}+c^{3}$   is
a) 7
b) 2
c) 0
d) -1

Explanation:

8. If a is a $3 \times3$  skew summertic matrix with real entries and trace of $A^{2}$ equals zero, then
a) A = O
b) 2A = I
c) A is orthogonal
d) none of these

Explanation:

9. If $\begin{bmatrix}i & 0 \\3 & -i \end{bmatrix}+X=\begin{bmatrix}i & 2 \\3 & 4+i \end{bmatrix}$      – X, then X is equal to
a) $\begin{bmatrix}0 & -1 \\3 & i \end{bmatrix}$
b) $\begin{bmatrix}0 & 1 \\0 & 2+i \end{bmatrix}$
c) $\begin{bmatrix}1 & 0 \\0 & 2-i \end{bmatrix}$
d) none of these

10. If $A=\begin{bmatrix}0 & -i \\i & 0 \end{bmatrix},B=\begin{bmatrix}1 & 0 \\0 & -1 \end{bmatrix}$     , then AB + BA is