Answer
See below
Work Step by Step
$A=\begin{bmatrix}
0 & 0 & 0\\ 1 & 0 & 0 \\ 0 & 1 & 0
\end{bmatrix} \\
A^2=\begin{bmatrix}
0 & 0 & 0\\ 1 & 0 & 0 \\ 0 & 1 & 0
\end{bmatrix} \begin{bmatrix}
0 & 0 & 0\\ 1 & 0 & 0 \\ 0 & 1 & 0
\end{bmatrix} =\begin{bmatrix}
0 & 0 & 0\\ 0 & 0 & 0 \\ 1 & 0 & 0
\end{bmatrix} $
$A^3=A^2.A=\begin{bmatrix}
0 & 0 & 0\\ 1 & 0 & 0 \\ 0 & 1 & 0
\end{bmatrix} \begin{bmatrix}
0 & 0 & 0\\ 0 & 0 & 0 \\ 1 & 0 & 0
\end{bmatrix} =\begin{bmatrix}
0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0
\end{bmatrix} $
Hence, an $n \times n$ matrix $A$ is nilpotent.
$e^{At}=I_3+At+\frac{1}{2!}(At)^2+\frac{1}{3!}(At)^3...+\frac{1}{k!}(At)^k\\
=\begin{bmatrix}
1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1
\end{bmatrix} +\begin{bmatrix}
-t & -6t & -5t\\ 0 & 0 & 0 \\ 0& t & 0
\end{bmatrix}+\frac{1}{2}\begin{bmatrix}
0 & 0 & 0\\ 0 & 0 & 0 \\\frac{1}{2}t^2 & 0 & 0
\end{bmatrix} \\
=\begin{bmatrix}
\begin{bmatrix}
1 & 0 & 0\\ t & 1 & 0 \\ \frac{1}{2}t^2 & t & 1
\end{bmatrix}
\end{bmatrix} $
