Conversion of the Permanent into the Determinant
P. M. Gibson · Proceedings of the American Mathematical Society · 1971
Let $A$ be an $n$-square $(0,1)$-matrix with positive permanent. It is shown that if the permanent of $A$ can be converted into a determinant by affixing $\pm$ signs to the elements of $A$ then $A$ has at most $({n^2} + 3n - 2)/2$ positive entries. Corollaries of this result are given.