Question

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

Answer 1

If B is a skew-symmetric matrix, then $B^T=-B$.

$$\begin{gathered} {\left(AB{A}^T\right)}^T={\left(A^T\right)}^T{B}^T{A}^T\left[\because {\left(ABC\right)}^T=C^T{B}^T{A}^T\right] \\ \Rightarrow {\left(AB{A}^T\right)}^T=A{B}^T{A}^T\left[\because {\left(A^T\right)}^T=A\right] \\ \Rightarrow {\left(AB{A}^T\right)}^T=A\left(-B\right)A^T\left[\because B^T=-B\right] \\ \Rightarrow {\left(AB{A}^T\right)}^T=-AB{A}^T \\ \therefore AB{A}^T\mathrm{is}\mathrm{a}\mathrm{skew}-\mathrm{symmetric}\mathrm{matrix}. \end{gathered}$$