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We derive closed form expressions for the mutual information of single input single output (SISO) and multiple input multiple output (MIMO) Rayleigh fading channels for any antenna number at any signal to noise ratio (SNR). Using these expressions, we show that the maximum mutual information of non\u2010coherent Rayleigh fading MIMO channels is achieved with a single transmitter and multiple receivers when the input distribution is Gaussian. We show that the addition of transmit antennas for a fixed number of receivers result in a reduction of mutual information. Furthermore, we argue that the mutual information is bounded by the SNR in both SISO and MIMO systems showing the sub\u2010optimality of Gaussian signalling in non\u2010coherent Rayleigh fading channels. 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