# Solve the Following Questions

Question:

Let $A=\left[\begin{array}{ll}2 & 3 \\ a & 0\end{array}\right], a \in \mathbf{R}$ be written as $P+Q$ where $P$ is a symmetric matrix and $Q$ is skew symmetric matrix. If $\operatorname{det}(Q)=9$, then the modulus of the sum of all possible values of determinant of $P$ is equal to:

1. 36

2. 24

3. 45

4. 18

Correct Option: 1

Solution:

$A=\left[\begin{array}{ll}2 & 3 \\ a & 0\end{array}\right], a \in R$

and $P=\frac{A+A^{T}}{2}=\left[\begin{array}{cc}2 & \frac{3+a}{2} \\ \frac{a+3}{2} & 0\end{array}\right]$

and $Q=\frac{A-A^{T}}{2}=\left[\begin{array}{cc}0 & \frac{3-a}{2} \\ \frac{a-3}{2} & 0\end{array}\right]$

As, $\operatorname{det}(Q)=9$

$\Rightarrow(a-3)^{2}=36$

$\Rightarrow a=3 \pm 6$

$\therefore \quad a=9,-3$

$=0-\frac{(\mathrm{a}-3)^{2}}{4}=0$, for $\mathrm{a}=-3$

$=0-\frac{(a-3)^{2}}{4}=-\frac{1}{4}(12)(12)$, for $a=9$

$\therefore$ Modulus of the sum of all possible values of $\operatorname{det}(\mathrm{P})=|-36|+|0|=36$ Ans.