Question

If $A$ is a symmetric matrix and $B$ is a skew- symmetrix matrix such that $A+B=\left[\begin{array}{cc}2& 3\\ 5& -1\end{array}\right],$ then $AB$ is equal to :

• A. $\left[\begin{array}{cc}-4& 2\\ 1& 4\end{array}\right]$
• B. $\left[\begin{array}{cc}4& -2\\ 1& -4\end{array}\right]$
• C. $\left[\begin{array}{cc}4& -2\\ -1& -4\end{array}\right]$
• D. $\left[\begin{array}{cc}-4& -2\\ -1& 4\end{array}\right]$

Answer 1

The correct option is C $\left[\begin{array}{cc}4& -2\\ -1& -4\end{array}\right]$

Given $A=A^T$ and $B=-B^T$
$\because A+B=\left[\begin{array}{cc}2& 3\\ 5& -1\end{array}\right]\dots \left(I\right)$
$(A+B{)}^T={\left[\begin{array}{cc}2& 3\\ 5& -1\end{array}\right]}^T=\left[\begin{array}{cc}2& 5\\ 3& -1\end{array}\right]$
$A^T+B^T=A-B=\left[\begin{array}{cc}2& 5\\ 3& -1\end{array}\right]\dots \left(II\right)$
Solving $\left(I\right)$ and $\left(II\right)$ we get
$A=\left[\begin{array}{cc}2& 4\\ 4& -1\end{array}\right]$ and $B=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$
$\therefore AB=\left[\begin{array}{cc}4& -2\\ -1& -4\end{array}\right]$