Calculate the Exponential of a Matrix

Question

Let $F_r(l)=e^{A_r(l)}=[f_{r,ij}]$, where $A_r(l)=\begin{bmatrix}-r&r\\-r&r-l\end{bmatrix}$, where $r>0,l>0$.

1. Calculate $F_r(l)$. (Analytic expression or approximation, in terms of series, if possible).
2. Show that $f_{r,12}+f_{r,21}=0$.

The original questions is the following: Let $A_r(A)=\begin{bmatrix} -rI&rI\\-rI&(r-1)I+A \end{bmatrix}$, where $A,I$ are of the same dimension and $I$ is the identity matrix. Calculate the coefficients for $A,A^2,A^3,\cdots$ in the expansion of $e^{A_r(A)}$.