Question

Find the symmetric and skew-symmetric matrices of matrix $A=\left[\begin{array}{ccc}0& -2& 4\\ 2& 0& -1\\ -4& 1& 0\end{array}\right]$.

Answer 1

$A=\left[\begin{array}{ccc}0& -2& 4\\ 2& 0& -1\\ -4& 1& 0\end{array}\right]$

$A^{\prime }=\left[\begin{array}{ccc}0& 2& -4\\ -2& 0& 1\\ 4& -1& 0\end{array}\right]=-\left[\begin{array}{ccc}0& -2& 4\\ 2& 0& -1\\ -4& 1& 0\end{array}\right]$

$=-A$

So $A$ is skew symmetric matrix

So, symmetric matrix of $A$ is $A$ itself and skew symmetric matrix from $A$ is zero matrix of order $3$.