Question

Write $A=\left[\begin{array}{cc}3& 5\\ 1& -1\end{array}\right]$ as the sum of a symmetric and a skew-symmetric matrix.

Answer 1

$A=\left[\begin{array}{cc}3& 5\\ 1& -1\end{array}\right],A^T=\left[\begin{array}{cc}3& 1\\ 5& -1\end{array}\right]P=\frac{1}{2}(A+A^T)=\frac{1}{2}\left[\begin{array}{cc}6& 6\\ 6& -2\end{array}\right]=\left[\begin{array}{cc}3& 3\\ 3& -1\end{array}\right]P^T=\left[\begin{array}{cc}3& 3\\ 3& -1\end{array}\right]$

Hence $P$ is a symmetric matrix

$A=\left[\begin{array}{cc}3& 5\\ 1& -1\end{array}\right],A^T=\left[\begin{array}{cc}3& 1\\ 5& -1\end{array}\right]Q=\frac{1}{2}(A-A^T)=\frac{1}{2}\left[\begin{array}{cc}0& 4\\ -4& 0\end{array}\right]=\left[\begin{array}{cc}0& 2\\ -2& 0\end{array}\right]Q^T=\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right]$

Hence $Q$ is a skew symmetric matrix

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