A noncommutative version of the matrix inversion formula

The cofactor expression (? 1)^i+j$\text{det}\text{(I ? B)}{}_{\mathrm{<mtext>j</mtext><mtext>?</mtext><mtext>i</mtext><mtext>?</mtext>}}\text{det}\text{(I ? B)}$ 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 Σ_m≥0B^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.