This paper investigates the achievable rate region of two-way amplify-and-forward (AF) relaying systems with multiple relays, where all nodes are equipped with multiple antennas. First, we find linear processing filters to optimize the weighted sum-rate for the two-way channels with sum power constraints. The proposed algorithm achieves the maximum weighted sum-rate by deriving the gradient expressions and iteratively applying the gradient descent method. Consequently, the proposed scheme outperforms the conventional methods in terms of the weighted sum-rate. Also, the achievable rate region is obtained by adjusting the weights in the proposed algorithm. It is observed from the ergodic rate region that channel reciprocity can improve the achievable sum-rate in two-way relay systems unlike one-way channels.