Question

If $A$ and $B$ are $n$ squared matrices then prove that $AB$ is symmetric if and only if $A$ and $B$ commute.

Answer 1

$A^{\prime }=A,B=B^{\prime }=B$ given as both $A$ and $B$ are symmetric. Let us choose that $A$ and $B$ are symmetric. Let us choose that $A$ and $B$ commute
i.e., $AB=BA$
$AB$ will be symmetric if $(AB{)}^{\prime }=AB$
Now $(AB{)}^{\prime }=B^{\prime }{A}^{\prime }=BA=AB$ by (1)
$\therefore$ $AB$ is symmetric
Converse: Choose that $AB$ is symmetric.
$\therefore$ $(AB{)}^{\prime }=AB$ or $B^{\prime }{A}^{\prime }=BA=AB$ or $BA=AB$
$\therefore$ $A$ and $B$ commute.