We consider the rate of convergence of the Markov chain Xn+1 = AXn + Bn (modp), where A is an integer matrix with nonzero eigenvalues, and {Bn}n is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of Qk invariant under A. If |λi | =1 for all eigenvalues λi of A, then n = O((ln p)2) steps are sufficient and n = O(ln p) steps are necessary to have Xn sampling from a nearly uniform distribution. Conversely, if A has the eigenvalues λi that are roots of positive integer numbers, |λ1| = 1 and |λi | > 1 for all i = 1, then O(p2) steps are necessary and sufficient.
