Question

If A and B are symmetric matrices, then write the condition for which AB is also symmetric.

Answer 1

Given: AB is symmetric.

$$\begin{gathered} \Rightarrow {\left(AB\right)}^T=AB \\ \Rightarrow B^T{A}^T=AB\left[\because {\left(AB\right)}^T=B^T{A}^T\right] \\ \Rightarrow BA=AB\left[\because A\mathrm{and}B\mathrm{are}\mathrm{symmetric}\mathrm{matrices},\mathrm{so}{A}^T=A\mathrm{and}{B}^T=B\right] \end{gathered}$$

Thus, AB is also symmetric, if AB = BA.