[Math] Fundamental matrix and exponential of matrix using Laplace Transform

Question

I'm trying to work out how to find $$\exp(At)$$ for a system of linear differential equations $$x'=Ax.$$

I know that the solution is a fundamental matrix of the system such that $$\exp(At)=I$$
at time $0$.

What is the method for solving this using the Laplace transform? The only method I can figure out is finding the eigenvalues and diagonalzing the matrix.

Answer 1

The Laplace transform of $exp(At)$ is $(sI-A)^{-1}$. So compute that latter and then take its inverse Laplace transform.