Let $A=\left[\begin{array}{cc}1 & 5 \\ -1 & 2\end{array}\right],$ then $A^{\prime}=\left[\begin{array}{cc}1 & -1 \\ 5 & 2\end{array}\right]$

Now $A+A^{\prime}=\left[\begin{array}{cc}1 & 5 \\ -1 & 2\end{array}\right]+\left[\begin{array}{cc}1 & -1 \\ 5 & 2\end{array}\right]=\left[\begin{array}{ll}2 & 4 \\ 4 & 4\end{array}\right]$

Let $P=\frac{1}{2}\left(A+A^{\prime}\right)=\left[\begin{array}{ll}1 & 2 \\ 2 & 2\end{array}\right]$

Now $P^{\prime}=\left[\begin{array}{ll}1 & 2 \\ 2 & 2\end{array}\right]=P$

Thus $P=\frac{1}{2}(A+A^{\prime})$ is symmetric matrix.

Now, $A-A^{\prime}=\left[\begin{array}{cc}1 & 5 \\ -1 & 2\end{array}\right]-\left[\begin{array}{cc}1 & -1 \\ 5 & 2\end{array}\right]=\left[\begin{array}{cc}0 & 6 \\ -6 & 0\end{array}\right]$

Let $Q=\frac{1}{2}\left(A-A^{\prime}\right)=\left[\begin{array}{cc}0 & 3 \\ -3 & 0\end{array}\right]$

Now $Q^{\prime}=\left[\begin{array}{cc}0 & -3 \\ 3 & 0\end{array}\right]=-Q$

Thus $Q=\frac{1}{2}(A-A^{\prime})$ is a skew symmetric matrix.

Representing $A$ as the sum of $P$ and $Q$.

$P+Q=\left[\begin{array}{ll}1 & 2 \\ 2 & 2\end{array}\right]+\left[\begin{array}{cc}0 & 3 \\ -3 & 0\end{array}\right]=\left[\begin{array}{cc}1 & 5 \\ -1 & 2\end{array}\right]=A$