In this paper we investigate the reconstruction conditions of nuclear norm minimisation for low-rank matrix recovery. We obtain a sufficient condition to guarantee the robust reconstruction and exact reconstruction of all r-rank matrices from linear measurements via nuclear norm minimisation. Furthermore, we not only show that our result could return to previous result, but also demonstrate that the gained upper bounds concerning the recovery error are better. Moreover, we prove that the restricted isometry property condition is sharp. Besides, the numerical experiments are conducted to reveal the nuclear norm minimisation method is stable and robust for the recovery of low-rank matrix.