  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.
