Question

Prove that +ive off integral powers of a skew asymmetric matrix are skew symmetric but +ive even integral powers are symmetric

Answer 1

Proceeding we can show that
$(A^p{)}^{\prime }=(A.A.A....ptimes{)}^{\prime }$
$=(A^{\prime }{A}^{\prime }{A}^{\prime }......ptimes)$
$={(-1)}^p{A}^p$
because $A^{\prime }=-A$ when $A$ is skew symmetric
$\therefore$ $(A^p{)}^{\prime }=A^p$ when $p$ is even and hence $A^p$ is symmetric
and $(A^p{)}^{\prime }=-A^p$ when $p$ is odd and hence $A^p$ is skew symmetric