In this work, density functional theory plane-wave pseudo potential method, with local density approximation (LDA) and generalized gradient approximation (GGA) are used to investigate the structural, elastic, mechanical and thermodynamic properties of the intermetallic compound Mg3Rh. Comparison of the calculated equilibrium lattice constants and experimental data shows very good agreement. The elastic constants were determined from a linear fit of the calculated stress-strain function according to Hooke's law. From the elastic constants, the bulk modulus B, shear modulus G, Young's modulus E, Poisson's ratio sigma, anisotropy factor A, the ratio B/G and the hardness parameter H for Mg3Rh compound are obtained. Our calculated elastic constants indicate that the ground state structure of Mg3Rh is mechanically stable. The calculation results show that this intermetallic crystal is stiff, elastically anisotropic and ductile material. The sound velocities and Debye temperature are also predicted from elastic constants. This is the first quantitative theoretical prediction of these properties. (C) 2016 Production and hosting by Elsevier B.V. on behalf of Chongqing University.