Advances in Mathematics, 31 (1979), p. 330-349.

Dominique Foata

A Noncommutative Version of the Matrix Inversion Formula

Abstract. The cofactor expression (-1)^(i+j) det(I_B)_(ĵî)/det(I-B) for the (i,j)-entry in the inverse of a matrix (I-B) is proved to be equal to the corresponding entry of the series ∑_(m≥0) B^m, by using purely combinatorial methods: circuit monoid techniques and monomial rearrangements. Moreover, the identity is shown to hold in a non-commutative formal power series algebra.

foata@unistra.fr

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