Given the matrix $(I-A)^{-1}$ and $B$, can we compute $e^{A+B}$, where $e^X$ is defined to be $\sum_{i=0}^{\infty} \frac{X^i}{i!}$.
(Note that $A$ and $B$ do not commute, and hence $e^A \cdot e^B \neq e^{A+B}$).
Now I've observed that Laplace transformation might be a useful tool. I've obtained that $\mathcal{L}[e^{tA+B}](s) ={(sI-A)}^{-1}e^{B}.$
So is the above (inverse) laplace transformation really useful to compute $e^{A+B}$ from $(I-A)^{-1}$ and $B$? How can I get the resultant $e^{A+B}$ from the Laplace transformation?
Hope anyone who is familiar with linear algebra and Laplace transformation could give me a hand. Thanks!