Question

If $A=\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]$, show that $A-A^T$ is a skew-symmetric matrix.

Answer 1

Given $A=\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]$.

Now

$A-A^T$

$=\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]$$-\left[\begin{array}{cc}2& 4\\ 3& 5\end{array}\right]$

$=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]=B$(let).

Now

$B^T$

$=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]=-B$.

So the matrix $A-A^T$ is skew symmetric.