a
(a) We have $A = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]$
$\therefore$ ${A^2} = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]\,\,\left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right] = {I_2}$
$\therefore$ ${A^4} = {A^2}.{A^2} = {I_2}.{I_2} = {I_2} = \left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right]$.