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

✍ Scribed by Pierre Lalonde

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 730 KB
• Volume: 158
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(95)00040-4

✦ Synopsis

We 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.