a
(a) ${A^2} = \left[ {\begin{array}{*{20}{c}}1&2\\0&1\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}1&2\\0&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&4\\0&1\end{array}} \right]$ and ${A^3} = {A^2}A$.
= $\left[ {\,\begin{array}{*{20}{c}}1&4\\0&1\end{array}\,} \right]\,\,\left[ {\,\begin{array}{*{20}{c}}1&2\\0&1\end{array}\,} \right] = \left[ {\,\begin{array}{*{20}{c}}1&6\\0&1\end{array}\,} \right]$ and so on.
$\therefore $ ${A^n} = \left[ {\,\begin{array}{*{20}{c}}1&{2n}\\0&1\end{array}\,} \right]$.