I am writing a dissertation on continuous time Markov's chains and I need some help to calculate the resolvent $R(\lambda)$ of the transition rate function $Q$ of a process. The resolvent is defined as the Laplace transform of the transition probability matrix $P(t)$, which is a matrix-function of time that is proved to be equal to the exponential matrix function $e^{Qt}$ using the Chapman-Kolmogorov equation.
The first doubt that I have is the following: How can I be sure that the Laplace transform of $e^{Qt}$ is defined?
Assuming that it is well defined, by applying the definition we get that the resolvent $R$ is equal to the following integral $$R(\lambda)=\int_{0}^{+\infty}e^{-\lambda t}e^{Qt}dt.$$ The ideal solution that I am expecting here is the following $$R(\lambda)=(\lambda I-Q)^{-1},$$ but to get to thet results I should be able to show that $e^{-\lambda t}$ $e^{Qt}=e^{-(\lambda I-Q)t}$. I tried to use the definition of the exponential of a matrix to get to that expression but it led to nothing. I also tried to use Cauchy's series product but I am not sure that it can be applied between seried that have different "dimension".