On the completion of a partial integral matrix to a unimodular matrix (Q869939)

A \textit{partial} \(n \times n\) integer matrix \(A\) is one in which some entries are indeterminates. A free diagonal for such a matrix is an \(n\)-tuple \(a_{1,\sigma(1)},\dots, a_{n,\sigma(n)}\) for a permutation \(\sigma\). The author shows that if \(A\) has a free diagonal, then there is an evaluation of the indeterminate entries that gives a unimodular matrix. Further, a partial matrix with \(2n-3\) prescribed entries so that any \(n\) do not constitute a row or column can also be completed to a unimodular matrix.