Let $F$ be the finite field of order $q$ and $\M(n,r, F)$ be the set of $n\times n$ matrices of rank $r$ over the field $F$. For $\alpha\in F$ and $A\in \M(n,F)$, let $$Z^{ \alpha}_{ A,r}=\left\{ X\in \M(n,r, F)\mid \tr(AX)=\alpha\right \}.$$ In this article, we solve the problem of determining the cardinality of $Z_{ A,r}^{ \alpha}$. We also solve the generalization of the problem to rectangular matrices.
Categories: math.RA