Question

If A is a skew-symmetric matrix and n is an even natural number, write whether Aⁿ is symmetric or skew-symmetric or neither of these two.

Answer 1

$$\begin{gathered} \mathrm{If}A\mathrm{is}\mathrm{a}\mathrm{skew}-\mathrm{symmetric}\mathrm{matrix},\mathrm{then}{A}^T=-A. \\ {\left(A^n\right)}^T={\left(A^T\right)}^n\left[\mathrm{For}\mathrm{all}n\in N\right] \\ \Rightarrow {\left(A^n\right)}^T={\left(-A\right)}^n\left[\because A^T=-A\right] \\ \Rightarrow {\left(A^n\right)}^T={\left(-1\right)}^n{A}^n \\ \Rightarrow {\left(A^n\right)}^T=A^n,\mathrm{if}n\mathrm{is}\mathrm{even}\mathrm{or}-A^n,\mathrm{if}n\mathrm{is}\mathrm{odd}. \end{gathered}$$

Hence, $A^n$ is symmetric when n is an even natural number.