Question

Express $\left[\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right]$ as a sum of symmetric and skew symmetric matrices.

Answer 1

Let, $A=\left[\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right]$

$⟹A^T=\left[\begin{array}{ccc}1& 2& 1\\ -1& 1& 1\\ 1& -3& 1\end{array}\right]$

Symmetric matrix $\frac{A+A^T}{2}=$$\frac{1}{2}\left[\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right]$$+\frac{1}{2}\left[\begin{array}{ccc}1& 2& 1\\ -1& 1& 1\\ 1& -3& 1\end{array}\right]$$=\frac{1}{2}\left[\begin{array}{ccc}2& 1& 2\\ 1& 2& -2\\ 2& -2& 2\end{array}\right]$

Skew - Symmetric matrix $\frac{A-A^T}{2}=$$\frac{1}{2}\left[\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right]$$-\frac{1}{2}\left[\begin{array}{ccc}1& 2& 1\\ -1& 1& 1\\ 1& -3& 1\end{array}\right]$$=\frac{1}{2}\left[\begin{array}{ccc}0& -3& 0\\ 3& 0& -4\\ 0& 4& 0\end{array}\right]$

$\therefore \left[\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right]$$=\frac{1}{2}\left[\begin{array}{ccc}2& 1& 2\\ 1& 2& -2\\ 2& -2& 2\end{array}\right]$$+\frac{1}{2}\left[\begin{array}{ccc}0& -3& 0\\ 3& 0& -4\\ 0& 4& 0\end{array}\right]$