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In this letter, the problem of determining the maximum weighted sum rate (WSR) for given arbitrary weights is investigated for the vector Gaussian broadcast channel (GBC). Although the solution to the problem in general involves an iterative algorithm, we focus on the important special case of the two-user channel to provide a closed-form expression for the maximum WSR. The achievable rate region is known to be identical for both the vector GBC and its dual uplink Gaussian multiple access channel (GMAC) due to the uplink-downlink duality. We first derive the optimal power allocation in the dual uplink channel, which is then applied to the capacity formula to obtain the desired result. The presented result is a generalization of the (unweighted) sum rate maximization of the two-user GBC provided by Caire and Shamai .