In this paper, we investigate an amplify-and-forward (AF) based two-way cooperative status update system, where two sources aim to exchange status updates with each other as timely as possible with the help of a relay. Specifically, the relay receives the sum signal from the two sources in one time slot, and then amplifies and forwards the received signal to both the sources in the next time slot. We adopt a recently proposed concept, the age of information (AoI), to characterize the timeliness of the status updates. Assuming that the two sources are able to generate status updates at the beginning of each time slot (i.e., generate-at-will model), we derive a closed-form expression of the expected weighted sum AoI of the considered system. We further minimize the expected weighted sum AoI by optimizing the transmission powers at all nodes under their individual peak power constraints. Simulation results corroborate the correctness of our theoretical analysis.