Summary: It is known that inverse M-matrices are strict path product (SPP) matrices, and that the converse is not true for matrices of order greater than 3. In this paper, given a normalized SPP-matrix A, some new values s $^{\prime}$ for which A + s $^{\prime}$ Iis an inverse M-matrix are obtained. Our values s $^{\prime}$ extend the values s given by Johnson and Smith [C.R. Johnson and R.L. Smith. Positive, path product, and inverse M-matrices. Linear Algebra Appl., 421:328-337, 2007.]. The question whether or not a $4 \times 4$ SPP-matrix is a P-matrix is settled.

15A48, 15A57

M-matrix, inverse M-matrix, path product matrix, P-matrix