The Stacks project

Lemma 11.6.2. Let $A$ be a finite central simple $k$-algebra. Any automorphism of $A$ is inner. In particular, any automorphism of $\text{Mat}(n \times n, k)$ is inner.

Proof. Note that $A$ is a finite central simple algebra over the center of $A$ which is a finite field extension of $k$, see Lemma 11.4.2. Hence the Skolem-Noether theorem (Theorem 11.6.1) applies. $\square$

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