Question

If A and B are symmetric matrices of the same order, write whether AB − BA is symmetric or skew-symmetric or neither of the two.

Answer 1

Since A and B are symmetric matrices, $A^T=A\mathrm{and}{B}^T=B$.

Here,
$$\begin{gathered} {\left(AB-BA\right)}^T={\left(AB\right)}^T-{\left(BA\right)}^T \\ \Rightarrow {\left(AB-BA\right)}^T=B^T{A}^T-A^T{B}^T\left[\because {\left(AB\right)}^T=B^T{A}^T\right] \\ \Rightarrow {\left(AB-BA\right)}^T=BA-AB\left[\because B^T=B\mathrm{and}{A}^T=A\right] \\ \Rightarrow {\left(AB-BA\right)}^T=-\left(AB-BA\right) \\ \mathrm{Therefore},AB-BA\mathrm{is}\mathrm{skew}-\mathrm{symmetric}. \end{gathered}$$