Each automorphism of the matrix algebra is inner.

Question

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.

Answer 1

For completeness: this is a special case of the Skolem-Noether theorem.

Answer 2

Let me develop the comments of Geoff Robinson further into an incomplete answer.
Since $M_n(k)$ has only one irreducible representation $V$, $V^{\alpha}$ and $V$ must be the same representation. Hence the corresponding matrices are conjugate: so $\alpha$ is inner, for $M_n(k)$ is semisimple.
Point out any error which occurs please. Thanks very much.