Question: Let,

\[A=\left[ \begin{matrix} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \\ \end{matrix} \right]\] show that
\[{{e}^{At}}={{e}^{2t}}\left[ \begin{matrix} 1 & t & \frac{{{t}^{2}}}{2} \\ 0 & 1 & t \\ 0 & 0 & 1 \\ \end{matrix} \right]\]