Answer
See below
Work Step by Step
$A=\begin{bmatrix}
-3 & 9\\ -1 & 3
\end{bmatrix} \\
A^2=\begin{bmatrix}
-3 & 9\\ -1 & 3
\end{bmatrix}\begin{bmatrix}
-3 & 9\\ -1 & 3
\end{bmatrix}=\begin{bmatrix}
0 & 0\\ 0 & 0
\end{bmatrix}$
Hence, an $n \times n$ matrix $A$ is nilpotent.
$e^{At}=I_2+At+\frac{1}{2!}(At)^2+...+\frac{1}{k!}(At)^k\\
=I_2+\begin{bmatrix}
-3t & 9t\\ -t & 3t
\end{bmatrix}+\frac{1}{2!}A^2t^2+...+\frac{1}{k!}A^kt^k\\
=\begin{bmatrix}
1 & 0\\ 0 & 1
\end{bmatrix}+\begin{bmatrix}
-3t & 9t\\ -t & 3t
\end{bmatrix}\\
=\begin{bmatrix}
1-3t & 9t\\ -t & 1+3t
\end{bmatrix}$
