Let $A = \left[ {\matrix{ 2 & 3 \cr a & 0 \cr } } \right]$, a$\in$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 determinant of P is equal to :

$A = \left[ {\matrix{ 2 & 3 \cr a & 0 \cr } } \right]$, ${A^T} = \left[ {\matrix{ 2 & a \cr 3 & 0 \cr } } \right]$<br/><br/>$A = {{A + {A^T}} \over 2} + {{A - {A^T}} \over 2}$<br/><br/>Let $P = {{A + {A^T}} \over 2}$ and $Q = {{A - {A^T}} \over 2}$<br/><br/>$$Q = \left( {\matrix{ 0 & {{{3 - a} \over 2}} \cr {{{a - 3} \over 2}} & 0 \cr } } \right)$$<br/><br/>Det (Q) = 9<br/><br/>$0 - \left( {{{3 - a} \over 2}} \right)\left( {{{a - 3} \over 2}} \right) = 9$<br/><br/>$$ \Rightarrow {\left( {{{a - 3} \over 2}} \right)^2} = 9 \Rightarrow {(a - 3)^2} = 36$$<br/><br/>$a - 3 = \pm \,6 \Rightarrow a = 9, - 3$<br/><br/>$$P = \left[ {\matrix{ 2 & {{{a + 3} \over 2}} \cr {{{a + 3} \over 2}} & 0 \cr } } \right]$$<br/><br/>$P = \left[ {\matrix{ 2 & 6 \cr 6 & 0 \cr } } \right]$ or $\left[ {\matrix{ 2 & 0 \cr 0 & 0 \cr } } \right]$ <br/><br/> | P | = - 36 or 0 <br/><br/>$\therefore$ | $-$36 + 0 | = 36