I'm working on the following problem and I'd like some guidance.
Describe up to isomorphism all semisimple $\mathbb{C}$-subalgebras of $M_4(\mathbb{C})$ (4 by 4 matrices over $\mathbb{C}$). Note that if $A$ and $B$ are $\mathbb{C}$-algebras and if $\alpha\in A$, and $\beta \in B$, have minimal polynomials $f,g$ respectively, then $(\alpha,\beta)\in A\oplus B$ has minimal polynomial $h = lcm(f,g)$
My thoughts are that by artin-wedderburn, this ring is simple because it's just a matrix ring, so shouldn't there should be no semi-simple subalgebras? but the hint seems to suggest that there are subalgebras.
By Wedderburn, any finite-dimensional semisimple algebra over $\mathbb C$ is of the form $$\tag1\bigoplus_{j=1}^m M_{n_j}(\mathbb C).$$ The question is how to fit this inside size $4\times 4$. In $(1)$ we would have $n_1+\cdots+n_m\leq 4$. So there are not that many choices: note that order is irrelevant, since isomorphism will not "see" the order of the blocks. So we may as well order the block sizes from bigger to smaller. Thus
$n_1=4$.
$n_1=3$, $n_2=1$
$n_1=2$, $n_2=2$
$n_1=2$, $n_2=1$, $n_3=1$
$n_1=n_2=n_3=n_4=1$
$n_1=3$
$n_1=2$, $n_2=1$
$n_1=n_2=n_3=1$
$n_1=n_2=1$
$n_1=1$