This paper concerns with the properties of Hadamard product of inverse M-matrices. Structures of tridiagonal inverse M-matrices and Hessenberg inverse M-matrices are analysed. It is proved that the product A AT satis es Willoughby's necessary conditions for being an inverse M-matrix when A is an irreducible inverse M-matrix. It is also proved that when A is either a Hessenberg inverse M-matrix or a tridiagonal inverse M-matrix then AAT is an inverse M-matrix. Based on these results, the conjecture that A?AT is an inverse M-matrix when A is an inverse M-matrix is made. Unfortunately, the conjecture is not true. Copyright