This paper describes a selection technique of an optimum random matrix using a genetic algorithm for speech recognition based on random projections. Random projections have been suggested as a means of dimensionality reduction, where the original data are projected onto a subspace using a random matrix. Moreover, as we are able to produce various random matrices, it may be possible to find a transform matrix that is superior to conventional transformation matrices among random matrices. In this paper, a genetic algorithm is introduced to find an optimum random matrix. Its effectiveness is confirmed by word recognition experiments.