Question

Write matrix $A$ as the sum of a symmetric and a skew symmetric matrix, where
$A=\left(\begin{array}{ccc}4& 2& -3\\ 1& 3& -6\\ -5& 0& -7\end{array}\right)$

Answer 1

We have
$A=\left(\begin{array}{ccc}4& 2& -3\\ 1& 3& -6\\ -5& 0& -7\end{array}\right)$
$\therefore A^{\prime }=\left(\begin{array}{ccc}4& 1& -5\\ 2& 3& 0\\ -3& -6& -7\end{array}\right)$
Then
$A+A^{\prime }=\left(\begin{array}{ccc}4& 2& -3\\ 1& 3& -6\\ -5& 0& 7\end{array}\right)+\left(\begin{array}{ccc}4& 1& -5\\ 2& 3& 0\\ -3& -6& -7\end{array}\right)=\left(\begin{array}{ccc}8& 3& -8\\ 3& 6& -6\\ -8& -6& 0\end{array}\right)....\left(1\right)$

$A-A^{\prime }=\left(\begin{array}{ccc}4& 2& -3\\ 1& 3& -6\\ -5& 0& 7\end{array}\right)-\left(\begin{array}{ccc}4& 1& -5\\ 2& 3& 0\\ -3& -6& -7\end{array}\right)=\left(\begin{array}{ccc}0& 1& 2\\ -1& 0& -6\\ -2& 6& 14\end{array}\right)....\left(2\right)$
Adding $\left(1\right)$ and $\left(2\right)$, we get
$2A=\left(\begin{array}{ccc}8& 3& -8\\ 3& 6& -6\\ -8& -6& 0\end{array}\right)+\left(\begin{array}{ccc}0& 1& 2\\ -1& 0& -6\\ -2& 6& 14\end{array}\right)$
$\therefore A=\left(\begin{array}{ccc}4& 3/2& -4\\ 3/2& 3& -3\\ -4& -3& 0\end{array}\right)+\left(\begin{array}{ccc}0& 1/2& 1\\ -1/2& 0& -3\\ -1& 3& 7\end{array}\right)$