Invertible sums of matrices

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.