  Let $r > 0$ be an integer, let $\mathbb{F}_q$ be a finite field of $q$
elements, and let $\mathcal{A}$ be a nonempty proper subset of $\mathbb{F}_q$.
Moreover, let $\mathbf{M}$ be a random $m \times n$ rank-$r$ matrix over
$\mathbb{F}_q$ taken with uniform distribution. We prove, in a precise sense,
that, as $m, n \to +\infty$ and $r,q,\mathcal{A}$ are fixed, the number of
entries of $\mathbf{M}$ that belong to $\mathcal{A}$ approaches a normal
distribution.
