  Let $R$ be a commutative ring and $M_n(R)$ be the ring of $n \times n$
matrices with entries from $R$. For each $S \subseteq M_n(R)$, we consider its
(generalized) null ideal $N(S)$, which is the set of all polynomials $f$ with
coefficients from $M_n(R)$ with the property that $f(A) = 0$ for all $A \in S$.
The set $S$ is said to be core if $N(S)$ is a two-sided ideal of $M_n(R)[x]$.
It is not known how common core sets are among all subsets of $M_n(R)$. We
study this problem for $2 \times 2$ matrices over $\mathbb{F}_q$, where
$\mathbb{F}_q$ is the finite field with $q$ elements. We provide exact counts
for the number of core subsets of each similarity class of $M_2(\mathbb{F}_q)$.
While not every subset of $M_2(\mathbb{F}_q)$ is core, we prove that as $q \to
\infty$, the probability that a subset of $M_2(\mathbb{F}_q)$ is core
approaches 1. Thus, asymptotically in~$q$, almost all subsets of
$M_2(\mathbb{F}_q)$ are core.
