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