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

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 all polynomials $f$ 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 ideal of 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 of which share the same irreducible characteristic polynomial. In particular, we show that if $F$ is finite with $q$ elements and $\mathrm{\mid }S\mathrm{\mid }⩾q^3-q^2+1$, then $S$ is core. As a byproduct of our work, we obtain some results on block Vandermonde matrices, invertible matrix commutators, and graphs defined via an invertible difference relation.