Symbiotic radio (SR) backscatter systems are possible techniques for the future low-power wireless communications for Internet of Things devices. In this paper, we propose a multiple-input-multiple-output (MIMO) SR backscatter system, where the secondary multi-antenna transmission from the backscatter device (BD) to the receiver is riding on the primary multi-antenna transmission from the transmitter to the receiver. We investigate the beamforming design optimization problem which maximizes the achievable rate of secondary transmission under the achievable rate constraint of primary transmission. In the MIMO SR backscatter system, each antenna of the SR BD reflects its received ambient radio frequency signals from all the transmitting antennas of the transmitter, which causes the globally optimal solution is difficult to obtain. In this paper, we propose a method to obtain the achievable rate upper bound. Furthermore, considering both primary and secondary transmissions, we propose an exact penalty method based locally optimal solution. Simulation results illustrate that our proposed exact penalty method based locally optimal solution performs close to the upper bound.