Question

If $A=\left[\begin{array}{ccc}1& 2& 3\\ 1& 1& 1\\ 2& 2& 4\end{array}\right]$ then prove that $A-A^T$ is skew symmetric matrix

Answer 1

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

${\therefore (A-A^T)}^T=-(A-A^T)$

$\therefore A-A^T$ ia a skew-symmetric matrix