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.