Combinatorial properties of some classes of matrices over GF(2)

We study several classes of matrices of GF(2) constructed from lists of subsets of finite sets In this paper. We show that all matrices in these classes are representations of connected equicardinal matrix over GF(2). In Matrix terms, these irreducible (defined below) matrices all have the property that every minimal dependent set of column has the same cardinality over GF(2). This fact is shown directly in this paper by elementary matrix considerations. In a subsequent paper, we shall show that these classes of matrices are in fact the classes of canonical forms for all representations of nontrivial binary connected equicardinal matroids.