Question

For any square matrix write whether AA^T is symmetric or skew-symmetric.

Answer 1

Here,

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

Thus, AA^T is a symmetric matrix.