  Given a positive integer $r$ and a prime power $q$, we estimate the
probability that the characteristic polynomial $f_{A}(t)$ of a random matrix
$A$ in $\mathrm{GL}_{n}(\mathbb{F}_{q})$ is square-free with $r$ (monic)
irreducible factors when $n$ is large. We also estimate the analogous
probability that $f_{A}(t)$ has $r$ irreducible factors counting with
multiplicity. In either case, the main term $(\log n)^{r-1}((r-1)!n)^{-1}$ and
the error term $O((\log n)^{r-2}n^{-1})$, whose implied constant only depends
on $r$ but not on $q$ nor $n$, coincide with the probability that a random
permutation on $n$ letters is a product of $r$ disjoint cycles. The main
ingredient of our proof is a recursion argument due to S. D. Cohen, which was
previously used to estimate the probability that a random degree $n$ monic
polynomial in $\mathbb{F}_{q}[t]$ is square-free with $r$ irreducible factors
and the analogous probability that the polynomial has $r$ irreducible factors
counting with multiplicity. We obtain our result by carefully modifying Cohen's
recursion argument in the matrix setting, using Reiner's theorem that counts
the number of $n \times n$ matrices with a fixed characteristic polynomial over
$\mathbb{F}_{q}$.
