Positive, path product, and inverse M-matrices

It is known that an inverse M-matrix is strict path product, but not necessarily vice versa for n > 3. It is shown that any square, positive matrix may be made strict path product by predictable additions to the diagonal and that any (normalized) strict path product matrix may be made inverse M by additions to the diagonal that are bounded in terms of the size of the matrix. The latter has implications for pairs of inverse M-matrices whose Hadamard product is inverse M. A determinantal inequality relating principal minors and certain associated almost principal minors is derived for normalized inverse M-matrices.