Almost-nonsingular entry pattern matrices

In an entry pattern matrix A, all entries are indeterminates and the same indeterminate may appear in multiple positions. For a field F, an F-completion of A results from assigning a value from F to each indeterminate entry. We say that a square entry pattern matrix is almost-nonsingular over a field F if all of its F-completions are nonsingular, except for those in which all indeterminates are assigned the same value. This work investigates bounds for the maximum number of indeterminates of almost-nonsingular entry pattern matrices over some fields, including the real field, the rational field and finite fields.