A noncommutative version of the matrix inversion formula

Abstract The cofactor expression (− 1)i+j det (I − B) j i 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 Σm≥0Bm, 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.