TITLE: Each automorphism of the matrix algebra is inner.
QUESTION [6 upvotes]: Exercise I.II.IV in the book Local Representation Theory by J.L.Alperin:  

Demonstrate that any automorphism of the algebra $M_n(k)$ is inner by using the fact that $M_n(k)$ has a unique simple module.  

I want only hints, as this seems elementary. Thanks for your attention.

REPLY [4 votes]: For completeness: this is a special case of the Skolem-Noether theorem.