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 $B^T=B$.

$$\begin{gathered} {\left(AB{A}^T\right)}^T={\left(A^T\right)}^T{B}^T{A}^T\left[\because {\left(ABC\right)}^T=C^T{B}^T{A}^T\right] \\ \Rightarrow {\left(AB{A}^T\right)}^T=A{B}^T{A}^T\left[\because {\left(A^T\right)}^T=A\right] \\ \Rightarrow {\left(AB{A}^T\right)}^T=AB{A}^T\left[\because B^T=B\right] \end{gathered}$$

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