Question

If A is a symmetric matrix and n ∈ N, write whether Aⁿ is symmetric or skew-symmetric or neither of these two.

Answer 1

$$\begin{gathered} \mathrm{If}A\mathrm{is}\mathrm{a}\mathrm{symmetric}\mathrm{matrix},\mathrm{then}{A}^T=A. \\ \mathrm{Now}, \\ {\left(A^n\right)}^T={\left(A^T\right)}^n\left[\mathrm{for}\mathrm{all}n\in N\right] \\ \Rightarrow {\left(A^n\right)}^T={\left(A\right)}^n\left[\because A^T=A\right] \end{gathered}$$

Hence, $A^n$ is a symmetric matrix.