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In this paper, we first derive an optimal precoder for amplify-and-forward (AF) multiple-antenna relay systems with single-antenna source and destination terminals by maximizing the output signal-to-noise ratio (SNR). Then, the resulting maximum SNR expression and its approximation at high transmission SNR are analyzed to obtain their statistical properties for independently identically distributed (i.i.d.) Rayleigh fading channels. Based on the properties of the approximate maximum SNR, the average symbol error probability (ASEP) is investigated, leading to an explicit diversity order and array gain. The ergodic achievable rate is also studied, giving a tight closed-form upper bound. The theoretical analysis is finally validated by Monte Carlo simulations.