Question

For the matrix $A=\left[\begin{array}{cc}1& 5\\ 6& 7\end{array}\right]$, verify that
(i) $(A+A^{\prime })$ is a symmetric matrix
(ii) $(A-A^{\prime })$ is a skew symmetric matrix.

Answer 1

$A^{\prime }=\left[\begin{array}{cc}1& 6\\ 5& 7\end{array}\right]$
(i) $A+A^{\prime }=\left[\begin{array}{cc}1& 5\\ 6& 7\end{array}\right]+\left[\begin{array}{cc}1& 6\\ 5& 7\end{array}\right]=\left[\begin{array}{cc}2& 11\\ 11& 14\end{array}\right]$
$\therefore {(A+A^{\prime })}^{\prime }=\left[\begin{array}{cc}2& 11\\ 11& 14\end{array}\right]=A+A^{\prime }$

Hence, $(A+A^{\prime })$ is a symmetric matrix.

(ii) $A-A^{\prime }=\left[\begin{array}{cc}1& 5\\ 6& 7\end{array}\right]-\left[\begin{array}{cc}1& 6\\ 5& 7\end{array}\right]=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$
$\therefore {(A-A^{\prime })}^{\prime }=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]=-\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]=-(A-A^{\prime })$

Hence, $(A-A^{\prime })$ is a skew symmetric matrix.