This letter presents a non-orthogonal multiple access (NOMA) technique for a two-cell multiple-input multiple-output (MIMO) system that exploits the alignments of inter-cell interference channels and signal channels within a cluster in a cell. The proposed technique finds combiner vectors for users that align the inter-cell interference channels and the signal channels simultaneously. This technique utilizes the aligned interference and signal channels to obtain precoder matrices for base stations through which each data stream modulated by NOMA can be transmitted to the intended cluster without interference. In addition, we derive the sufficient condition for transmit and receive antenna configurations in the MIMO NOMA systems to eliminate inter-cell interference and inter-cluster interference simultaneously. Because the proposed technique effectively suppresses the inter-cell interference, it achieves a higher degree of freedom than the existing techniques relying on an avoidance of inter-cell interference, thereby obtaining a better sum rate performance in high SNR regions. Furthermore, we present the hybrid MIMO NOMA technique, which combines the MIMO NOMA technique exploiting channel alignment with the existing techniques boosting the received signal powers. Using the benefits from these techniques, the proposed hybrid technique achieves a good performance within all SNR regions. The simulation results successfully demonstrate the effectiveness of the proposed techniques on the sum rate performance.