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In this letter, the multiple-input multiple-output (MIMO) system with a precoder is considered in the transmit-correlated Rayleigh channels. We especially target the MIMO system employing zero forcing receivers. Based on random matrix theory, we first derive the exact probability density function (PDF) of the signal-to-interference-plus-noise ratio (SINR). Using the derived SINR PDF and a close approximation of the Gaussian Q-function, we derive a tight closed-form approximation of the symbol error rate (SER). In the high signal-to-noise ratio (SNR) regime, we also propose a high-SNR power allocation (HPA) by minimizing the global SER approximation under the total power constraint. Our SER analysis suggests that the SER approximation can be used to estimate the error probability. At high SNRs, the computationally efficient HPA converging to the optimal PA achieves noticeable performance gain over the equal PA, particularly in the high transmit correlation scenario. Furthermore, the performance gain of the proposed HPA increases with the diversity order.