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In this paper, a novel reconfigurable intelligent surface (RIS)-assisted multiple-input multiple-output (MIMO) symbiotic radio (SR) system is proposed, in which an RIS, operating as a secondary transmitter (STx), sends messages to a multi-antenna secondary receiver (SRx) by using cognitive backscattering communication, and simultaneously, it enhances the primary transmission from a multi-antenna primary transmitter (PTx) to a multi-antenna primary receiver (PRx) by intelligently reconfiguring the wireless environment. We are interested in the joint design of active transmit beamformer at the PTx and passive reflecting beamformer at the STx to minimize the total transmit power at the PTx, subject to the signal-to-noise-ratio (SNR) constraint for the secondary transmission and the rate constraint for the primary transmission. Due to the non-convexity of the formulated problem, we decouple the original problem into a series of subproblems using the alternating optimization method and then iteratively solve them. The convergence performance and computational complexity of the proposed algorithm are analyzed. Furthermore, we develop a low-complexity algorithm to design the reflecting beamformer by solving a backscatter link enhancement problem through the semi-definite relaxation (SDR) technique. Then, theoretical analysis is performed to reveal the insights of the proposed system. Finally, simulation results are presented to validate the effectiveness of the proposed algorithms and the superiority of the proposed system.