If $A$, $B$ are square matrices of same order and $B$ is a skew-symmetric matrix, then show that ${A^\prime }BA$ is skew-symmetric.

Since, $A$ and $B$ are square matrices of same order and $B$ is a skew-symmetric matrix i.e.,

${B^\prime } = - B$.
Now, we have to prove that ${A^\prime }BA$ is a skew-symmetric matrix.

$\therefore$ ${A^\prime }B{A^\prime } = {A^\prime }B{A^\prime } = B{A^\prime }{A^\prime }$

$= {A^\prime }{B^\prime }A = {A^\prime } - BA = - {A^\prime }BA$

Hence, ${A^\prime }BA$ is a skew-symmetric matrix.

LONG ANSWER