## A fact about matrix exponentials

Let $\lambda \in \mathbb{C}$ and let $M$ be an $n \times n$ matrix over $\mathbb{C}$. Prove that $\mathsf{exp}(\lambda I + M) = e^{\lambda}\mathsf{exp}(M)$.

Note that $\lambda I$ and $M$ commute. Using two previous exercises (here and here), we have $\mathsf{exp}(\lambda I + M) = \mathsf{exp}(\lambda I)\mathsf{exp}(M)$ $= \mathsf{exp}(\lambda)I \mathsf{exp}(M)$ $= e^\lambda \mathsf{exp}(M)$.