Show that ${A^\prime }A$ and $A{A^\prime }$ are both symmetric matrices for any matrix A.

Let $P = {A^\prime }A$
$\therefore$ ${P^\prime } = {\left( {A{A^\prime }} \right)^\prime }$

$= {A^\prime }{\left( {{A^\prime }} \right)^\prime }$

$= {A^\prime }A = P$

Therefore, ${A^\prime }A$ is symmetric matrix for any matrix A.

Similarly, let $Q = A{A^\prime }$

$\therefore$ ${Q^\prime } = {\left( {A{A^\prime }} \right)^\prime } = {\left( {{A^\prime }} \right)^\prime }{(A)^\prime }$

$= A{\left( {{A^\prime }} \right)^\prime } = Q$

Therefore, $A{A^\prime }$ is symmetric matrix for any matrix A.