${A^2} = \left[ {\begin{array}{*{20}{c}}
  0 \\ 
  2 
\end{array}\,\,\,\,\begin{array}{*{20}{c}}
  4 \\ 
  1 
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  0 \\ 
  2 
\end{array}\,\,\,\,\begin{array}{*{20}{c}}
  4 \\ 
  1 
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  8 \\ 
  2 
\end{array}\,\,\,\,\begin{array}{*{20}{c}}
  4 \\ 
  9 
\end{array}} \right]$

$\left. \begin{gathered}
  {A^2} = \left[ {\begin{array}{*{20}{c}}
  0 \\ 
  2 
\end{array}\,\,\,\,\begin{array}{*{20}{c}}
  4 \\ 
  1 
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  0 \\ 
  2 
\end{array}\,\,\,\,\begin{array}{*{20}{c}}
  4 \\ 
  1 
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  8 \\ 
  2 
\end{array}\,\,\,\,\begin{array}{*{20}{c}}
  4 \\ 
  9 
\end{array}} \right] \hfill \\
  mA + nI = \left[ {\begin{array}{*{20}{c}}
  0 \\ 
  {2m} 
\end{array}\,\,\,\,\begin{array}{*{20}{c}}
  {4m} \\ 
  m 
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
  n \\ 
  0 
\end{array}\,\,\,\begin{array}{*{20}{c}}
  0 \\ 
  n 
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  n \\ 
  {2m} 
\end{array}\,\,\,\,\,\begin{array}{*{20}{c}}
  {4m} \\ 
  {m + n} 
\end{array}} \right] \hfill \\ 
\end{gathered}  \right\} \Rightarrow n = 8\,\,\,,\,\,\,m = 1$