Past exam of the mathematical methods I course of the University of Oxford / 2016 resit / 10 / c / Solution

In a submultiplicative matrix norm, the exponential series is absolutely convergent. Its Cauchy product can therefore be grouped by the total degree:When $AB=BA$, the inner sum is $(A+B{)}^n$ by the binomial theorem. Thus $\boxed{e^A{e}^B=e^{A+B}}$. The commutation assumption is used precisely in that binomial expansion; absolute convergence alone does not supply it.