A non-commutative version of Jacobi's equality on the cofactors of a matrix

AbstractWe give a combinatorial proof of Jacobi's equality relating a cofactor of a matrix with the complementary cofactor of its inverse. This result unifies two previous approaches of the combinatorial interpretation of determinants: generating functions of weighted permutations and generating functions of families (configurations) of non-crossing paths. We show that Jacobi's equality is valid with the same choice of non-commutative entries as in Foata's proof of matrix inversion by cofactors