If $A = \left[ {\begin{array}{llllllllllllllllllll}0&1\\1&0\end{array}} \right]$, then ${A^2}$ is equal to

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

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