  For 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 $k^{-2+\epsilon}$ for the probability that $M$ is singular, and
size $k^{-1+\epsilon}$ 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.
