For the matrix $A=\left[\begin{array}{ll}1 & 5 \\ 6 & 7\end{array}\right]$, verify that
(i) $\left(A+A^{\prime}\right)$ is a symmetric matrix
(ii) $\left(A-A^{\prime}\right)$ is a skew symmetric matrix
$A^{\prime}=\left[\begin{array}{ll}1 & 6 \\ 5 & 7\end{array}\right]$
(i) $A+A^{\prime}$$=\left[\begin{array}{ll}1 & 5 \\ 6 & 7\end{array}\right]+\left[\begin{array}{ll}1 & 6 \\ 5 & 7\end{array}\right]=\left[\begin{array}{ll}2 & 11 \\ 11 & 14\end{array}\right]$
$\therefore\left(A+A^{\prime}\right)^{\prime}=\left[\begin{array}{ll}2 & 11 \\ 11 & 14\end{array}\right]=A+A^{\prime}$
Hence, $\left(A+A^{\prime}\right)$ is a symmetric matrix.
(ii) $A-A^{\prime}$$=\left[\begin{array}{ll}1 & 5 \\ 6 & 7\end{array}\right]-\left[\begin{array}{ll}1 & 6 \\ 5 & 7\end{array}\right]=\left[\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right]$
$\left(A-A^{\prime}\right)$$=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]=-\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]=-\left(A-A^{\prime}\right)$
Hence, $\left(A-A^{\prime}\right)$ is a skew-symmetric matrix.
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