Answer
See below
Work Step by Step
$A=\begin{bmatrix} A_0 & 0 \\ 0 & B_0 \end{bmatrix}$ is a block diagonal matrix with diagonal block matrices $A_0$ and $B_0$.
We have:
$e^{At}=I_n+At+\frac{(At)^2}{2!}+...+\frac{(At)^k}{k!}+...$
Obtain:
$A^2=\begin{bmatrix} A_0 & 0 \\ 0 & B_0 \end{bmatrix}\begin{bmatrix} A_0 & 0 \\ 0 & B_0 \end{bmatrix}=\begin{bmatrix} A_0^2 & 0 \\ 0 & B_0^2 \end{bmatrix}$
$A^n=\begin{bmatrix} A_0^n & 0 \\ 0 & B_0^n \end{bmatrix}$
then $e^{At}=\begin{bmatrix} I & 0 \\ 0 & I \end{bmatrix}+\begin{bmatrix} A_0t & 0 \\ 0 & B_0t \end{bmatrix}+\begin{bmatrix} \frac{A_0^2t^2}{2} & 0 \\ 0 & \frac{B_0^2t^2}{2} \end{bmatrix}+...\begin{bmatrix} \frac{A_0^kt^k}{k!} & 0 \\ 0 & \frac{B_0^kt^k}{k!} \end{bmatrix}\\
=\begin{bmatrix} I_0+A_0t+\frac{A_0^2t^2}{2}+... \frac{A_0^kt^k}{k!}...+0 \\ 0 +...I_0+B_0t+\frac{A_0^2t^2}{2}+... \frac{B_0^kt^k}{k!}...0 \end{bmatrix}\\
=\begin{bmatrix} e^{A_0t} & 0 \\ 0 & e^{B_0t} \end{bmatrix}$
