## Matrices and Determinants Questions and Answers Part-3

1. If $A\left(\theta\right)=\left(\begin{array}{c}1 & \tan\theta\\ -\tan\theta & 1\end{array}\right)$     and AB = I, then $\left(sec^{2}\theta\right)B$   is equal to
a) $A\left(\theta\right)$
b) $A\left(-\theta\right)$
c) $A\left(\theta/2\right)$
d) $A\left(-\theta/2\right)$

Explanation: As AB = I, we get B = A-1

2. if $A=\begin{bmatrix}a & b \\b & a \end{bmatrix}$    and $A^{2}=\begin{bmatrix}\alpha & \beta \\\beta & \alpha \end{bmatrix}$     then
a) $\alpha =a^{2}+b^{2},\beta=2ab$
b) $\alpha =a^{2}+b^{2},\beta=a^{2}-b^{2}$
c) $\alpha =2ab ,\beta=a^{2}+b^{2}$
d) $\alpha =a^{2}+b^{2},\beta=ab$

Explanation:

3. If $A=\begin{bmatrix}a & b \\c & d \end{bmatrix}$     is such that |A| = 0 and $A^{2}-\left(a+d\right)A+kI=O$    , then k is equal to
a) b+c
b) a+d
c) ab+cd
d) 0

Explanation: As |A| = 0, we get ad – bc = 0

4. If $A=\begin{bmatrix}\alpha & 2 \\2 & \alpha \end{bmatrix}$     and $\mid A^{3}\mid =125$    , then $\alpha$ is equal to
a) $\pm 3$
b) $\pm 2$
c) $\pm 5$
d) 0

Explanation: We have 125 = |A3| = |A|3

5. If $A=\begin{bmatrix}a+ib & c+id \\-c+id & a-ib \end{bmatrix}$     and $a^{2}+b^{2}+c^{2}+d^{2}=1$     , then $A^{-1}$ is equal to
a) $\begin{bmatrix}a+ib & -c+id \\-c+id & a-ib \end{bmatrix}$
b) $\begin{bmatrix}a-ib & c-id \\-c-id & a+ib \end{bmatrix}$
c) $\begin{bmatrix}a-ib & -c-id \\c-id & a+ib \end{bmatrix}$
d) none of these

Explanation:

6. The number of values of k for which the system of equations (k + 1)x + 8y = 4k, kx + (k + 3)y = 3k – 1 has no solution is
a) 0
b) 3
c) 2
d) 1

Explanation: For the system of equations to have no solution, we must have

7. if $\begin{bmatrix}\cos\left(\pi/6\right) & \sin\left(\pi/6\right) \\-\sin\left(\pi/6\right) & \cos\left(\pi/6\right) \end{bmatrix}A=\begin{bmatrix}1 & 1 \\0 & 1 \end{bmatrix}$
and Q=PAP' then $P'Q^{2019}P$   is equal to
a) $\begin{bmatrix}1 & \sqrt{3}/2 \\0 & 2019 \end{bmatrix}$
b) $\begin{bmatrix}1 & 2019 \\0 & 1 \end{bmatrix}$
c) $\begin{bmatrix}\sqrt{3}/2 & 2019 \\0 & 1 \end{bmatrix}$
d) $\begin{bmatrix}\sqrt{3}/2 & -1/2 \\1 & 2019 \end{bmatrix}$

Explanation:

8.If $A=\begin{bmatrix}a_{1} & b_{1} & c_{1} \\a_{2} & b_{2} & c_{2} \\a_{3} & b_{3} & c_{3}\end{bmatrix}$     and $\mid A\mid \neq 0$  , then the system of equations $a_{1}x+b_{1}y+c_{1}z=0 , a_{2}x+b_{2}y+c_{2}z=0$         and $a_{3}x+b_{3}y+c_{3}z=0$     has
a) only one solution
b) infinite number of solutions
c) no solution
d) More than one but finite number of solution

Explanation: Note that A is invertible as |A| $\neq$ 0.

9. If $\left(\begin{array}{c}1 & -\tan\theta\\ \tan\theta & 1\end{array}\right)\left(\begin{array}{c}1 & \tan\theta\\ -\tan\theta & 1\end{array}\right)^{-1}=\begin{bmatrix}a & -b \\b & a \end{bmatrix}$
then
a) a = b = 1
b) $a=\cos 2\theta, b=\sin 2\theta$
c) $a=\sin 2\theta, b=\cos 2\theta$
d) $a=1, b=\sin 2\theta$

10. Let $S_{k}=\left(\begin{array}{c}1 & k\\ 0 & 1\end{array}\right),k\epsilon N$       then $\left(S_{2}\right)^{n}\left(S_{k}\right)^{-1}$    (where $n\epsilon N$) is equal to:
a) $S_{2n+k}$
b) $S_{2n-k}$
c) $S_{2^{n}+k-1}$
d) $S_{2^{n}-k}$