The asymptotic probability that a random biased matrix is invertible

Abstract Let q = pe be a power of a prime. Suppose we are given a probability distribution on GF(q) not concentrated on any proper affine subspace of GF(q) regarded as a vector space over its prime subfield GF(p). Let M be a random n by n matrix whose entries are chosen independently from the given distribution and let An be the probability that M is invertible. We show that lim n →∞ A n = Π i=1 ∞ 1− 1 q i .