If matrix $A = {\left[ {{a_{ij}}} \right]_{2 \times 2}}$, where ${a_{ij}} = 1$, if $i \ne j = 0$ and if $i = j$, then ${A^2}$ is equal to

We have, $A = {\left[ {{a_{ij}}} \right]_{2 \times 2}}$, where ${a_{ij}} = 1$, if $i \ne j = 0$ and if $i = j$

$\therefore$ $A = \left[ {\begin{array}{llllllllllllllllllll}0&1\\1&0\end{array}} \right]$

and ${A^2} = \left[ {\begin{array}{llllllllllllllllllll}0&1\\1&0\end{array}} \right]\left[ {\begin{array}{llllllllllllllllllll}0&1\\1&0\end{array}} \right] = \left[ {\begin{array}{llllllllllllllllllll}1&0\\0&1\end{array}} \right] = {\rm{I}}$