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Let $F$ be a field and $M_n(F)$ the ring of $n \times n$ matrices over $F$.Given a subset $S$ of $M_n(F)$, the null ideal of $S$ is the set of allpolynomials $f$ with coefficients from $M_n(F)$ such that $f(A) = 0$ for all $A\in S$. We say that $S$ is core if the null ideal of $S$ is a two-sided idealof the polynomial ring $M_n(F)[x]$. We study sufficient conditions under which$S$ is core in the case where $S$ consists of $3 \times 3$ matrices, all ofwhich share the same irreducible characteristic polynomial. In particular, weshow that if $F$ is finite with $q$ elements and $|S| \geqslant q^3-q^2+1$,then $S$ is core. As a byproduct of our work, we obtain some results on blockVandermonde matrices, invertible matrix commutators, and graphs defined via aninvertible difference relation.

Null ideals of sets of $3 \times 3$ similar matrices with irreducible characteristic polynomial