Sylvester's Determinant Identity

$$\vert A\vert\,\vert A_{rs,pq}\vert=\vert A_{r,p}\vert\,\vert A_{s,q}\vert-\vert A_{r,q}\vert\,\vert A_{s,p}\vert,$$

where $A_{u,w}$ is the submatrix of ${\hbox{\sf A}}$ formed by the intersection of the subset $w$ of columns and $u$ of rows.