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

Let $F$ be the finite field of order $q$ and $\text{M}(n,r,F)$ be the set of $n\times n$ matrices of rank $r$ over the field $F$. For $\alpha \in F$ and $A\in \text{M}(n,F)$, let $$Z_{A,r}^{\alpha }=\{X\in \text{M}(n,r,F)\mid \text{tr}(AX)=\alpha \}\mathrm{.}$$ 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.