Question

If a is A symmetric matrix and B is a skew symmetric 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 $\left[\begin{array}{cc}4& -2\\ -1& -4\end{array}\right]$

$A=A^{\prime },B=-B^{\prime }$

$A+B=\left[\begin{array}{cc}2& 3\\ 5& -1\end{array}\right]$ $\left(1\right)$

$A^{\prime }+B^{\prime }=\left[\begin{array}{cc}2& 5\\ 3& -1\end{array}\right]$

$A-B=\left[\begin{array}{cc}2& 5\\ 3& -1\end{array}\right]$ $\left(2\right)$

After adding Eq. $\left(1\right)$ & $\left(2\right)$

$A=\left[\begin{array}{cc}2& 4\\ 4& -1\end{array}\right],B=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$

$AB=\left[\begin{array}{cc}4& -2\\ -1& -4\end{array}\right]$.