On the cardinality of matrices with prescribed rank and partial trace over a finite field

Abstract: 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.