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 Σ 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.
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