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Extensive studies have been conducted on the rate analysis of downlink multiuser multiple-input-multiple-output (MIMO) systems. Most of them, nevertheless, focused on the sum rate of all the users where the effect of large-scale fading is either ignored or simplified. For interference-limited MIMO cellular networks, users in various locations may achieve drastically different rates due to distinct large-scale fading and interference levels. It is, therefore, of great practical interest to characterize the rate performance of each single user under various power allocation strategies and precoding schemes. This paper presents an asymptotic per-user rate analysis for a 1-tier (i.e., 7-cell) downlink MIMO cellular network with M base-station antennas co-located at the center of each cell and K uniformly distributed users each equipped with N antennas. Explicit expressions of the ergodic rate with two representative linear precoding schemes, singular-value-decomposition (SVD) transmission and block diagonalization (BD), are derived, based on which the effect of power allocation strategies is further evaluated. The analysis shows that the rate performance of each user is sensitive to its position even if the average received power is kept a constant. To achieve a uniform rate all over the cell, an interference-aware power allocation scheme is proposed, with which the transmit power allocated to each user is carefully adjusted according to the large-scale fading coefficients from the user to its own and neighboring base stations. The tradeoff between fairness and sum rate performance is also identified, which highlights the importance of the characterization of per-user rate.