In this paper, the full-potential, linearized augmented plane wave (FP-LAPW) method was employed in investigating full-Heusler Co2CrA1’s structural, elastic, magnetic and electronic properties. The FP-LAPW method was employed in computing the structural parameters (bulk modulus, lattice parameters, c/a and first pressure derivatives). The optimized structural parameters were determined by generalized gradient approximation (GGA) for the exchange-correlation potential, Vxc. Estimating the energy gaps for these compounds was accomplished through modified Becke–Johnson potential (mBJ). It was found that the conventional Heusler compound Co2CrA1 with mBJ and CGA approaches had a half-metallic character, and its spin-down configuration had an energy gap. It was also found that the conventional and inverse Heusler Cr2MnSb and tetragonal (139) (Co2CrA1, Cr2MnSb) compounds with a half-metallic character had direct energy gaps in the spin-down configuration. To a certain degree, the total magnetic moments for the two compounds were compatible with the theoretical and experimental results already attained. Mechanically, we found that the conventional and inverse full-Heusler compound Co2CrAl was stable, but the inverse Cr2MnSb was unstable in the ferromagnetic state. The conventional Heusler compound Cr2MnSb was mechanically stable in the ferromagnetic state.