Recently, spin-gapless semiconductors (SGSs) and half-metallic materials (HMMs) have received considerable interest in the fields of materials sciences and solid-state physics because they can provide a high degree of spin polarization in electron transport. The results on band structure calculations reveal that the metallic fully-compensated ferrimagnet (M-FCF) Mn2ZnMg becomes half-metallic fully-compensated ferrimagnet (HM-FCF), fully-compensated ferrimagnetic semiconductor (FCF-S) and fully-compensated ferrimagnetic spin-gapless semiconductor (FCF-SGS) if the uniform strain applied. However, the metallic fully-compensated ferrimagnetism property of the Mn2ZnMg is robust to the tetragonalization. The structure stability based on the calculations of the cohesion energy and the formation energy of this compound has been tested. Furthermore, a magnetic state transition from antiferromagentic (AFM) state to non-magnetic (NM) state can be observed at the lattice constant of 5.20 Å.