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).

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