Counting matrices over finite rank multiplicative groups

Motivated by recent works on statistics of matrices over sets of number theoretic interest, we study matrices with entries from arbitrary finite subsets $\mathcal{A}$ of finite rank multiplicative groups infields of characteristic zero. We obtain upper bounds, in terms of the size of $\mathcal{A}$, on the number of such matrices of a given rank, with a given determinant and with a prescribed characteristic polynomial. In particular, in the case of ranks, our results can be viewed as a statistical version of work by Alon and Solymosi (2003).