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$Z_{A,r}^{\alpha }=\{X\in M(n,r,F\left)\right|\mathrm{Tr}\left(AX\right)=\alpha \}.$ 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.