A non-commutative version of Jacobiʹs equality on the cofactors of a matrix Original Research Article

We give a combinatorial proof of Jacobiʹs 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 Jacobiʹs equality is valid with the same choice of non-commutative entries as in Foataʹs proof of matrix inversion by cofactors.