Let (X i) be a sequence of m × m i.i.d. stochastic matrices with distribution μ. Then μ n is the distribution of X n X n−1 ...X 1. Simple sufficient conditions for the weak convergence of (μ n ) are presented here. An extremely simple (and verifiable) necessary and sufficient condition is provided for m= 3. The method for m= 3 works for m> 3 even though calculations are more involved for higher values of m. We also discuss the purity of the limit distribution for m≥2.