Distributional properties of powers of matrices

We apply the larger sieve to bound the number of 2×2 matrices not having large order when reduced modulo the primes in an interval. Our motivation is the relation with linear recursive congruential generators. Basically our results establish that the probability of finding a matrix with large order modulo many primes drops drastically when a certain threshold involving the number of primes and the order is exceeded. We also study, for a given prime and a matrix, the existence of nearby non-similar matrices having large order. In this direction we find matrices of large order when the trace is restricted to take values in a short interval.