Jacobi's identity and the König-Egerváry theorem
The König-Egerváry Theorem states that the maximum size of a partial matching in a relation is equal to the minimum size of a separating set of lines (a set of rows or columns such that every nonzero entry of the associated adjacency matrix is contained in a row or column of the set). The following weaker version of Jacobi's identity is used to prove the above theorem: let \(\Pi\) be a nonsingular matrix with rows and columns indexed by the set I. If M,\(N\subset I\) and \(| M| =| N|\), then \(\Pi\) (\(M| N)\) is nonsingular iff \(adj \Pi(M^ c| N^ c)\) is nonsingular.
scientific article; zbMATH DE number 3859135