Does the exponential of a matrix commute with the matrix?

Question

Can someone give me an idea for the proof that for every $t\in \mathbb{C}$ we have $e^{tA}\cdot A = A \cdot e^{tA} =$ ? I couldn't find a counterexample, so my gues is, that it would be true, but I'm not sure even how to begin the proof.

Answer 1

$$e^{tA}\cdot A = \left(\sum_{k=0}^\infty \frac{t^kA^k}{k!}\right)\cdot A$$ $$= \sum_{k=0}^\infty \frac{t^kA^{k+1}}{k!}$$ $$= A \cdot \left(\sum_{k=0}^\infty \frac{t^kA^k}{k!}\right)$$ $$= A \cdot e^{tA}$$