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 (n,r, F) be the set of n× n matrices of rank r over the field F. For α∈ F and A∈ (n,F), let ZαA,r=\X∈ (n,r, F) (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.
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