Invertible sums of matrices

TL;DR

This paper presents an elementary proof of a Caratheodory-type theorem on matrix invertibility, deriving determinant identities and extending results to commutative rings, local rings, and semilocal rings.

Contribution

It provides a new elementary proof of invertibility criteria and determinant identities, extending classical results to broader algebraic structures.

Findings

Determinant of large matrix sums expressed via smaller sums

Stabilization of determinant ideals in filtered families

Characterization of local and semilocal rings using matrix invertibility

Abstract

We give an elementary proof of a Caratheodory-type result on the invertibility of a sum of matrices, due first to Facchini and Barioli. The proof yields a polynomial identity, expressing the determinant of a large sum of matrices in terms of determinants of smaller sums. Interpreting these results over an arbitrary commutative ring gives a stabilization result for a filtered family of ideals of determinants. Generalizing in another direction gives a characterization of local rings. An analogous result for semilocal rings is also given -- interestingly, the semilocal case reduces to the case of matrices.