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

TL;DR

This paper determines the number of matrices with a given rank and prescribed trace condition over finite fields, extending the results to rectangular matrices and providing explicit cardinality formulas.

Contribution

It provides explicit formulas for counting matrices with fixed rank and trace over finite fields, including a generalization to rectangular matrices.

Findings

Derived formulas for the cardinality of matrices with prescribed rank and trace.

Extended results to rectangular matrices.

Solved the enumeration problem over finite fields.

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.