Question

Let $A$ being a square matrix then prove that $A-A^T$ is a skew-symmetric matrix.

Answer 1

Let $B=A-A^T$.

Now,

$B^T$

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

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

$=A^T-A$ [ Since $(A^T{)}^T=A$]

$=-B$.

So $B^T=-B$.

So, $A-A^T$ is a skew-symmetric matrix.