A matrix in a set M of matrices is prime (naturally enough) if it is not the product of any other matrices in the set. We thought we would look for the prime matrices in the set M of all 2 x 2 matrices with entries in the nonnegative integers and with determinant 1. To our great surprise we discovered that:
THEOREM. In M the matrices
are the only primes, and any member of M (except I) can be uniquely factorised into a product of those two.