Characterization of full rank ACI-matrices over fields

An ACI-matrix over a field FF is a matrix whose entries are polynomials with coefficients on FF, the degree of these polynomials is at most one in a number of indeterminates, and where no indeterminate appears in two different columns. In 2011 Huang and Zhan characterized the m×nm×n ACI-matrices such that all its completions have rank equal to min{m,n}min{m,n} whenever |F|⩾max{m,n+1}|F|⩾max{m,n+1}. We will give a characterization for arbitrary fields by introducing two classes of ACI-matrices: the maximal and the minimal full rank ACI-matrices.