Nonsingular ACI-matrices over integral domains

An ACI-matrix is a matrix whose entries are polynomials of degree at most one in a number of indeterminates where no indeterminate appears in two different columns. Consider the next two problems: (a) characterize the ACI-matrices of order n all of whose completions have the same nonzero constant determinant; (b) characterize the ACI-matrices of order n all of whose completions are nonsingular. In 2010 Brualdi, Huang and Zhan solved both problems for fields of at least n+1 elements. We extend their result on problem (a) to integral domains, and extend their result on problem (b) to arbitrary fields.