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In this paper, capacity performance analysis is presented for multiple-input multiple-output (MIMO) relay channels with transmit antenna selection and maximal-ratio combining receive (TAS/MRC) in amplify-and-forward (AF) relay networks operating over flat Rayleigh fading channels. Assuming that the source and destination are equipped with Ns and Nj antennas, respectively, and communicate with each other with the help of a single-antenna relay, we derive the cumulative distribution function (CDF), probability density function (PDF) and moment generation function (MGF) for the system end-to-end SNR. Based on these, we then present closed-form analytical expressions for the calculation of the ergodic capacity and outage capacity for the AF MIMO relay channels with TAS/MRC. Numerical results show that all the derived analytical expressions are very tight and are valid for evaluations of the ergodic and outage capacity under various antenna configurations. The results also show how the number of transmit antennas and receives antennas affect the overall channel capacity under fading channel environments.