Answer
See below
Work Step by Step
$A=\begin{bmatrix}
-1 & -6 & -5\\ 0 & -2 & -1 \\ 1 & 2 & 3
\end{bmatrix} \\
A^2=\begin{bmatrix}
-1 & -6 & -5\\ 0 & -2 & -1 \\ 1 & 2 & 3
\end{bmatrix} \begin{bmatrix}
-1 & -6 & -5\\ 0 & -2 & -1 \\ 1 & 2 & 3
\end{bmatrix}=\begin{bmatrix}
-4 & 8 & -4\\ -1 & -2 & -1 \\ 2 & -4 & 2
\end{bmatrix} $
$A^3=A^2.A=\begin{bmatrix}
-4 & 8 & -4\\ -1 & 2 & -1 \\ 2 & -4 & 2
\end{bmatrix} \begin{bmatrix}
-1 & -6 & -5\\ 0 & -2 & -1 \\ 1 & 2 & 3
\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 & -2t & -t \\ t & 2t & 3t
\end{bmatrix}+\frac{1}{2}\begin{bmatrix}
-4t^2 & 8t^2 & -4t^2\\ -t^2 & 2t^20 & -t^2 \\ 2t^2 & -4t^2 & 2t^2
\end{bmatrix} \\
=\begin{bmatrix}
1-t-4t^2 & -6t+8t^2 & -5t-4t^2\\ -t^2 & 1-2t-2t^2 & -t-t^2 \\ t+2t^2 & 2t-4t^2 & 1+3t+2t^2
\end{bmatrix} $
