  For $n \geq 225$ we show that every integer of the form $n + 2m$ such that $0
\leq 2m \leq n^{2} - \frac{9}{2} n \sqrt{n}$ is the dimension of a connected
semi-simple subalgebra of $\mathrm{M}_{n}(k)$, that is, a subalgebra isomorphic
to a direct sum of $t$ disjoint subalgebras $\mathrm{M}_{n_{i}}(k)$, where
$\sum_{i=1}^{t} n_{i} = n$. From this, we conclude that the density of integers
in $[0,\ldots, n^{2}]$ which are the dimension of a semi-simple subalgebra of
$\mathrm{M}_{n}(k)$ tends to $1$ as $n \rightarrow \infty$.
