Each automorphism of the matrix algebra is inner.
For completeness: this is a special case of the Skolem-Noether theorem.
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.