Given some $f$ that follows the setup defined in Laplace Transform we have that $$\mathscr{L}(\dot{f})=s\mathscr{L}(f)-f(0)$$

Let $A\in M_{n}(\mathbb{R})$ and suppose for some $s\in\mathbb{C}$ we have $$\mathrm{Re}(s)>\max_{i}\mathrm{Re}(\lambda_{i})$$ where $\lambda_{i}$ is the $i$th eigenvalue of $A$. Then we have that the Laplace Transform of the matrix exponential of $A$ is as follows: $$\mathscr{L}(e^{At})=(sI-A)^{-1}$$