  In this paper, we study the combinatorial relations between the cokernels
$\text{cok}(A_n+px_iI_n)$ ($1 \le i \le m$) where $A_n$ is an $n \times n$
matrix over the ring of $p$-adic integers $\mathbb{Z}_p$, $I_n$ is the $n
\times n$ identity matrix and $x_1, \cdots, x_m$ are elements of $
\mathbb{Z}_p$ whose reductions modulo $p$ are distinct. For a positive integer
$m \le 4$ and given $x_1, \cdots, x_m \in \mathbb{Z}_p$, we determine the set
of $m$-tuples of finitely generated $\mathbb{Z}_p$-modules $(H_1, \cdots, H_m)$
for which $(\text{cok}(A_n+px_1I_n), \cdots, \text{cok}(A_n+px_mI_n)) = (H_1,
\cdots, H_m)$ for some matrix $A_n$. We also prove that if $A_n$ is an $n
\times n$ Haar random matrix over $\mathbb{Z}_p$ for each positive integer $n$,
then the joint distribution of $\text{cok}(A_n+px_iI_n)$ ($1 \le i \le m$)
converges as $n \rightarrow \infty$.
