If A is a symmetric matrix and $\mathrm{n}\in \mathrm{N} \mathrm{then} {\mathrm{A}}^{\mathrm{n}}$ is

$$\begin{gathered} {\mathrm{A}}^{\mathrm{T}}=\mathrm{A}, {\left({\mathrm{A}}^{\mathrm{n}}\right)}^{\mathrm{T}}={\left({\mathrm{A}}^{\mathrm{T}}\right)}^{\mathrm{n}} \\ ={\mathrm{A}}^{\mathrm{n}}, \mathrm{for} \mathrm{n}\in \mathrm{N} \end{gathered}$$

$\therefore {\mathrm{A}}^{\mathrm{n}}$ is a symmetric matrix.