Addressing issues such as unbalanced DC-side capacitor voltage of the nine bridge arms and sub-module (SM) capacitor voltage fluctuations during the operation of the modular multilevel matrix converter (M3C), a circulating current control strategy for bridge arm voltage balancing is proposed. A mathematical decoupling model for both the input and output sides of the M3C is established. Through the analysis of the power of each bridge arm in the M3C, a hierarchical control strategy for bridge arm capacitor voltage balancing is designed, eliminating severe oscillations caused by low-frequency AC components of capacitor voltage ripple under the conditions f_m≈0 and f_m=f_g. A semi-physical experimental system for the M3C is built on the RT-LAB platform, and system operation under normal, low-frequency, and special operating conditions is simulated. Experimental results show that the proposed bridge arm voltage balancing control strategy performs well, with small fluctuation range of DC-side capacitor voltage and better output power quality.