2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/144544For a fixed $n\ge2$, consider an $n\times n$ matrix $M$ whose entries are random integers bounded by $k$ in absolute value. In this paper, we examine the probability that $M$ is singular (hence has eigenvalue 0), and the probability that $M$ has at least one rational eigenvalue. We show that both of these probabilities tend to 0 as $k$ increases. More precisely, we establish an upper bound of size $k^{-2+ε}$ for the probability that $M$ is singular, and size $k^{-1+ε}$ for the probability that $M$ has a rational eigenvalue. These results generalize earlier work by Kowalsky for the case $n=2$ and answer a question posed by Hetzel, Liew, and Morrison.9 pages, 1 figureNumber Theory15A36, 15A52 (Primary) 11C20, 15A18, 60C05 (Secondary)text