Question

Write a square matrix which is both symmetric as well as skew-symmetric.

Answer 1

$$\begin{gathered} \mathrm{Let}A=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ {A}^T=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \mathrm{Since}{A}^T=A,A\mathrm{is}\mathrm{a}\mathrm{symmmetric}\mathrm{matrix}. \\ \mathrm{Now}, \\ -A=-\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \Rightarrow -A=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \mathrm{Since}{A}^T=-A,\hspace{0.17em}A\mathrm{is}a\mathrm{skew}-\mathrm{symmetric}\mathrm{matrix}. \\ \mathrm{Thus},A=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\mathrm{is}\mathrm{an}\mathrm{example}\mathrm{of}\mathrm{a}\mathrm{matrix}\mathrm{that}\mathrm{is}\mathrm{both}\mathrm{symmetric}\mathrm{and}\mathrm{skew}-\mathrm{symmetric}. \end{gathered}$$