I was reading a proof of the Cayley-Hamilton theorem here, in which the author used matrices in power series. I found out that one can write
$$\left( \frac{1}{m} I + A \right)^{-1}$$
as a power series (the Neumann series) and that it converges under certain conditions. What I don't understand is why one can write $$\mbox{adj}(I-tA)$$ as a power series. I didn't find any information about writing matrices in general as power series and I didn't encounter anything like that before in my studies. Why
$$\mbox{adj}(I-tA) = \sum_{i=0}^\infty B_it^i$$
and, more generally, under what conditions can one write matrices as power series and how?