Let $\mathrm{A}=\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right]$. If M and N are two matrices given by $M=\sum _{k=1}^{10}{A}^{2k}$ and $N=\sum _{k=1}^{10}{A}^{2k-1}$ then ${\mathrm{MN}}^2$ is

$$\begin{gathered} A=\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right] \\ \Rightarrow {\mathrm{A}}^2=\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right]\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right]=\left[\begin{array}{cc}-4& 0\\ 0& -4\end{array}\right]=-4\mathrm{I} \\ \therefore {\mathrm{A}}^3=-4\mathrm{A} \\ \Rightarrow {\mathrm{A}}^4=(-4\mathrm{I})(-4\mathrm{I})=(-4{)}^2\mathrm{I} \\ \therefore {\mathrm{A}}^5=(-4{)}^2\mathrm{A}, {\mathrm{A}}^6=(-4{)}^3\mathrm{I} \\ \mathrm{M}=\sum _{\mathrm{k}=1}^{10}{\mathrm{A}}^{2\mathrm{k}}={\mathrm{A}}^2+{\mathrm{A}}^4+\dots .+{\mathrm{A}}^{20} \\ =\left[-4+(-4{)}^2+(-4{)}^3+\dots .+(-4{)}^{20}\right]\mathrm{I} \\ =\mathrm{\lambda A}\Rightarrow \text{ skew symmetric } \\ \Rightarrow {\mathrm{N}}^2\text{ is symmetric matrix } \\ \Rightarrow {\mathrm{MN}}^2\text{ is non identity symmetric matrix } \end{gathered}$$