Representation of $*$-automorphism on finite dimensional matrix algebras

As an alternative to Robin Chapman's solution, I would like to state Exercise 7.8 from Rørdam's, Larsen's and Laustsen's "Introduction to the K-theory of C*-algebras":

For every unital AF-algebra $A$ there is a short exact sequence
$$
1tooverline{mathrm{Inn}}(A)tomathrm{Aut}(A)tomathrm{Aut}(K_0(A))to 1,
$$
where $overline{mathrm{Inn}}(A)$ denotes approximately inner automorphisms and $mathrm{Aut}(K_0(A))$ denotes group automorphisms preserving the unit class and the positive cone in $K_0(A)$.

If $A$ is the matrix ring, then $mathrm{Aut}(K_0(A))$ is trivial and hence every automorphism of $A$ is approximately inner. Since $A$ is separable, every approximately inner automorphism is the pointwise limit of a sequence of inner automorphisms. And I think the finite-dimensionality of $A$ implies that the pointwise limit of a sequence of inner automorphisms is again inner.

Using the statement above, one immediately sees that, for instance, $mathbb Coplusmathbb C $ possesses an automorphism which is not approximately inner.