The Range of the Determinant Function on the Set of \(n x n\) \((0, 1)\)-Matrices

As stated in \([2]\), there is a conjecture that the determinant function maps the set of \(n \times n\) \((0, 1)\)-matrices onto a set of consecutive integers for any given \(n\). While this is true for \(n \leq 6\), it is shown to be false for \(n = 7\). In particular, there is no \(7 \times 7\) determinant in the range \(28 – 31\), but there is one equal to \(32\). Then the more general question of the range of the determinant function for all \(n\) is discussed. A lower bound is given on the largest string of consecutive integers centered at \(0\), each of which is a determinant of an \(n \times n\) \((0, 1)\)-matrix.