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It is shown that, if for some m , 2 ≤ m < n all principal submatrices of an n -by- n P -matrix A are monotone, then A is monotone, generalizing a known result that also assumed symmetry. It is also shown that the P -matrix assumption cannot be entirely eliminated. However, in addition, A is doubly monotone (n = m − 1 above, without the P -matrix assumption) if and only if A − 1 is invertible path product. It was known that inverse M -matrices are path product (they are more than doubly monotone). But (invertible) path product matrices are more general, and this is the first full and functional characterization of them. Several examples are given to illustrate the results and their generality. [ABSTRACT FROM AUTHOR]