Rectangular submatrices of inverse -matrices and the decomposition of a positive matrix as a sum

We characterize those square partial matrices whose specified entries constitute a rectangular submatrix that may be completed to an inverse -matrix. Together with the notion of an interior inverse -matrix, this is used to show that any positive matrix is a sum of inverse -matrices and to estimate the number of summands needed to represent a given matrix. Nonnegative matrices are also considered. There are substantial differences from the analogous problem of decomposing a positive matrix as a sum of totally positive matrices. In particular, the upper bound on the number of inverse -matrix summands is much less than that in the totally positive case (although an example is given to show that the number of totally positive summands may be less than the required number of inverse -matrix summands).