  Let ${\rm M}(V)={\rm M}(n,\mathbb{F}_q)$ denote the algebra of $n\times n$
matrices over $\mathbb{F}_q$, and let ${\rm M}(V)_U$ denote the (maximal
reducible) subalgebra that normalizes a given $r$-dimensional subspace $U$ of
$V=\mathbb{F}_q^n$ where $0<r<n$. We prove that the density of non-cyclic
matrices in ${\rm M}(V)_U$ is at least $q^{-2}\left(1+c_1q^{-1}\right)$, and at
most $q^{-2}\left(1+c_2q^{-1}\right)$, where $c_1$ and $c_2$ are constants
independent of $n,r$, and $q$. The constants $c_1=-\frac43$ and
$c_2=\frac{35}3$ suffice.
