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×n matrices of rank r over the field F. For α∈F and A∈M(n,F), letZA,rα={X∈M(n,r,F)|Tr(AX)=α}. In this article, we solve the problem of determining the cardinality of ZA,rα. We also solve the generalization of the problem to rectangular matrices.