Question

If $A$ and $B$ are symmetric matrices of the same order, then Show that $AB$ is symmetric if and only if $AB=BA$.

Answer 1

Given $A^1=A;B^1=B$
Let $AB$ be symmetric then ${\left(AB\right)}^1=AB\Rightarrow B^1{A}^1=AB\Rightarrow BA=AB$
Conversely if $AB=BA$ then
${\left(AB\right)}^1=B^1{A}^1=BA$

Hence, $AB=AB$