Let $n\in\mathbb{N}$, let $S_n$ denote the symmetric group on $\{1,\dots,n\}$, let $\operatorname{sgn}:S_n\to\{1,-1\}$ denote the signature map, and let $\det:\mathbb{R}^{n\times n}\to\mathbb{R}$ denote the standard determinant. For every matrix $A=(A_{ij})_{1\le i,j\le n}\in\mathbb{R}^{n\times n}$,