Question

If $B$ is skew symmetric matrix write whether the matrix $\left(AB{A}^T\right)$ is symmetric or skew symmetric.

Answer 1

Given $B$ is a skew symmetric matrix,

$⟹B^T=-B$,

Now,

$A{B}^{}{A}^T=\left(AB\right)A^T$,

Consider,

${\left(\left(AB\right)A^T\right)}^T=A{\left(AB\right)}^T=A{B}^T{A}^T=-AB{A}^T$

Therefore it is a skew symmetric matrix.