  An element $X$ in the algebra ${\rm M}(n,\mathbb{F})$ of all $n\times n$
matrices over a field $\mathbb{F}$ is said to be $f$-cyclic if the underlying
vector space considered as an $\mathbb{F}[X]$-module has at least one cyclic
primary component. These are the matrices considered to be `good' in the
Holt-Rees version of Norton's irreducibility test in the MeatAxe algorithm. We
prove that, for any finite field $\mathbb{F}_q$, the proportion of matrices in
${\rm M}(n,\mathbb{F}_q)$ that are `not good' decays exponentially to zero as
the dimension $n$ approaches infinity. Turning this around, we prove that the
density of `good' matrices in ${\rm M}(n,\mathbb{F}_q)$ for the MeatAxe depends
on the degree, showing that it is at least
$1-\frac2q(\frac{1}{q}+\frac{1}{q^2}+\frac{2}{q^3})^n$ for $q\geq4$. We
conjecture that the density is at least
$1-\frac1q(\frac{1}{q}+\frac{1}{2q^2})^n$ for all $q$ and $n$, and confirm this
conjecture for dimensions $n\leq 37$. Finally we give a one-sided Monte Carlo
algorithm called IsfCyclic to test whether a matrix is `good', at a cost of
${\rm O}({\rm Mat}(n)\log n)$ field operations, where ${\rm Mat}(n)$ is an
upper bound for the number of field operations required to multiply two
matrices in ${\rm M}(n,\mathbb{F}_q)$.
