We use the first-principles-based density functional theory with full potential linearized augmented plane wave method in order to investigate the structural, elastic, electronic, magnetic and thermoelectric properties of RhCrZ (Z= Si, Ge, P, As) Half-Heusler compounds. The preferred configurations of the RhCrZ alloys are all type a. The structural parameters are in good agreement with the available theoretical results. The Young’s and shear modulus, Poisson’s ratio, sound velocities, Debye temperature and melting temperature have been calculated. Furthermore, the elastic constants Cij and the related elastic moduli confirm their stability in the cubic phase and demonstrate their ductile nature. The compounds RhCrSi, RhCrGe, RhCrP and RhCrAs are found to be half-metallic ferrimagnets (HMFs) with a half-metallic gap EHM of 0.37, 0.35, 0.25 and 0.02 eV, respectively. The half-metallicity of RhCrZ (Z= Si, Ge, P, As) compounds can be kept in a quite large hydrostatic strain and tetragonal distortion. The Curie temperatures of RhCrSi, RhCrGe, RhCrP and RhCrAs compounds are estimated to be 952, 1261, 82 and 297 K, respectively, in the mean field approximation (MFA). Thermoelectric properties of the RhCrZ (Z= Si, Ge, P, As) materials are additionally computed over an extensive variety of temperatures and it is discovered that RhCrAs demonstrates higher figure of merit than RhCrSi, RhCrGe and RhCrP. The properties of half-metallicity and higher Seebeck coefficient make this material a promising candidate for thermoelectric and spintronic device applications