In this paper, we study the joint distribution of the cokernels of random p-adic matrices. Let p be a prime and P1(t), • • • , P l (t) ∈ Zp[t] be monic polynomials whose reductions modulo p in Fp[t] are distinct and irreducible. We determine the limit of the joint distribution of the cokernels cok(P1(A)), • • • , cok(P l (A)) for a random n × n matrix A over Zp with respect to Haar measure as n → ∞. By applying the linearization of a random matrix model, we also provide a conjecture which generalizes this result. Finally, we provide a sufficient condition that the cokernels cok(A) and cok(A + Bn) become independent as n → ∞, where Bn is a fixed n × n matrix over Zp for each n and A is a random n × n matrix over Zp.