Question

If B is a symmetric matrix, write whether the matrix AB A^T is symmetric or skew-symmetric.

Answer 1

If B is a symmetric matrix, then
$$B^T=B$$
${\displaystyle {(AB A^T)}^T={\left(A^T\right)}^T{B}^T{A}^T {[\because ABC)}^T=C^T{B}^T{A}^T]}$
${\displaystyle {(AB A^T)}^T=A{B}^T{A}^T [\because {\left(A^T\right)}^T=A]}$
${\displaystyle {(AB A^T)}^T=AB A^T [\because B^T=B]}$

 

 ${\displaystyle \therefore AB A^T}$ is a symmetric matrix .