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Mechanical and magnetic properties of Rh in bcc, fcc structures and RhH in cubic structure are investigated by using first-principles calculations. Theoretical strengths of these structures are given for the first time. The results show that the NaCl-type cubic RhH has a lower bulk modulus and a theoretical strength larger than those of Rh in bcc and fcc structures. A strong magneto-volume effect of a transition from low magnetic moment-low cell volume to high magnetic moment-large cell volume is also found for both the bcc and fcc Rh structures as well as RhH.