Question

Let A and B be two symmetric matrices. prove that AB=BA if and only if AB is a symmetric matrix.

Answer 1

Given:$A$ and$B$ are symmetric matrix

$A^T=A,B^T=B$

We need to show $AB$ is symmetric if and only if $A$ and $B$ commute.

and if $A$ and $B$ commute $(AB=BA)$, then $AB$ is symmetric.

Part:$I$

If $AB$ is symmetric then $A$ and $B$ commute.

Given $AB$ is symmetric

$\Rightarrow {\left(AB\right)}^T=AB$

$\Rightarrow B^T{A}^T=AB$ using the property ${\left(AB\right)}^T={\left(BA\right)}^T$

$\Rightarrow BA=AB$ (given$A^T=A$ and $B^T=B$)

Hence $A$ and $B$ commute.

Hence proved.

Part:$II$If $A$ and $B$ commute, then $AB$ is symmetric.

Given $A$ and $B$ commute.

i.e., $AB=BA$

We need to show $AB$ is symmetric.

We need to show ${\left(AB\right)}^T=B^T{A}^T$ as

${\left(AB\right)}^T=B^T{A}^T$

$=BA$ as $A=A^T$ and $B=B^T$ given

$=AB$ assumed that $AB=BA$

So, ${\left(AB\right)}^T=AB$

Hence, $AB$ is symmetric.

Hence proved.

Hence,$AB$ is symmetric if and only if $A$ and $B$ commute, that is $AB=BA$