Let A be a random n by n matrix over Z_p with respect to the Haar measure. Friedman and Washington proved that the distribution of the cokernel of A follows the Cohen-Lenstra distribution. In this talk, we introduce two possible ways to generalize their work. In particular we calculate the joint distribution of the cokernels cok(P_1(A)), ... , cok(P_l(A)) for polynomials P_1(t), ... , P_l(t)∈Z_p[t] under some mild conditions. We also provide a way to understand the linearization of a random matrix model using our result.