Prior works on the performance analysis of multi-antenna adaptive modulation systems have focused on the rich scattering environments. However, in practice, the transmitted signal may be subject to insufficient scattering or keyhole effect. A multiple-input multiple-output (MIMO) channel model that captures the effect of the double scattering is the Rayleigh product channel. In this paper, we present an analytic study of the adaptive modulation scheme for transmit beamforming (TBF) systems in Rayleigh product MIMO channels. In order to adjust the modulation constellation size, the signal-to-noise ratio (SNR) at the maximum ratio combining (MRC) receiver output is sent back to the transmitter. The modulation order is then fixed with respect to the channel conditions. Analytical expressions of system performance, such as average spectral efficiency (SE), bit error rate (BER), and outage probability (OP), are derived and confirmed by Monte Carlo simulations.