Let $\mathbb{ F}_q$ be the finite field of order $q$, and $\mathcal{ A}$ a non-empty proper subset of $\mathbb{ F}_q$. Let $\mathbf{ M}$ be a random $m \times n$ matrix of rank $r$ over $\mathbb{ F}_q$ taken with uniform distribution. It was proved recently by Sanna that as $m,n \to \infty$ and $r,q,\mathcal{ A}$ are fixed, the number of entries of $\mathbf{ M}$ in $\mathcal{ A}$ approaches a normal distribution. The question was raised as to whether or not one can still obtain a central limit theorem of some sort when $r$ goes to infinity in a way controlled by $m$ and $n$. In this paper we answer this question affirmatively.
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