Question

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

• A. $24$
• B. $18$
• C. $45$
• D. $36$

Answer 1

The correct option is D $36$

$Q=\frac{A-A^T}{2}=\frac{\left[\begin{array}{cc}2& 3\\ a& 0\end{array}\right]-\left[\begin{array}{cc}2& a\\ 3& 0\end{array}\right]}{2}\Rightarrow Q=\frac{1}{2}\left[\begin{array}{cc}0& 3-a\\ a-3& 0\end{array}\right]\Rightarrow det\left(Q\right)=\frac{1}{4}(a-3{)}^2=9\Rightarrow a-3=\pm 6\Rightarrow a=9,-3$

$P=\frac{A+A^T}{2}=\frac{1}{2}\left[\begin{array}{cc}4& 3+a\\ a+3& 0\end{array}\right]\Rightarrow det\left(P\right)=-\frac{1}{4}(a+3{)}^2$
$\Rightarrow det\left(P\right)=0$ or $\frac{-144}{4}=-36$
$\therefore$ Required sum $=|0-36|=36$