Let $\mathrm{A}=\left[\begin{array}{cc}2& 3\\ \mathrm{a}& 0\end{array}\right], \mathrm{a}\in \mathrm{R}$ be written as P + Q where P is a symmetric matrix and Q is skew symmetric matrix if det Q = 9 then the modulus of  the sum of all possible values of determi- nant of P is equal to :

$\begin{array}{rcl}\mathrm{P}+\mathrm{Q}& =& \left[\begin{array}{cc}2& 3\\ \mathrm{a}& 0\end{array}\right]\\ \mathrm{P}& =& \frac{1}{2}(\mathrm{A}+{\mathrm{A}}^{\mathrm{T}})=\frac{1}{2}\left[\begin{array}{cc}4& 3+\mathrm{a}\\ 3+\mathrm{a}& 0\end{array}\right]\\ \mathrm{Q}& =& \frac{1}{2}(\mathrm{A}-{\mathrm{A}}^{\mathrm{T}})=\frac{1}{2}\left[\begin{array}{cc}0& 3-\mathrm{a}\\ \mathrm{a}-3& 0\end{array}\right]\\ \left|\mathrm{Q}\right|& =& 9\\ & \Rightarrow & \frac{1}{4}(0+{(\mathrm{a}-3)}^2)=9\Rightarrow {(\mathrm{a}-3)}^2=36\Rightarrow \mathrm{a}-3=\pm 6 \\ \mathrm{a}& =& 9, -3\\ \therefore \left|\mathrm{P}\right|& =& 0 \mathrm{for} \mathrm{a}=-3\\ & =& -36 \mathrm{for} \mathrm{a}=9\\ \therefore \mathrm{Sum} \mathrm{of} \mathrm{values} \mathrm{of} \left|\mathrm{P}\right|& =& \left|-36\right|+0\\ & =& 36\\ & & \end{array}$