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Consider the matrix

$\begin{pmatrix} -12 & 6 &0\\ 58 & 34 & 18 \\ 18 & 12 & 6\end{pmatrix}$

The problem ask me to decompose the kernel and cokernel of this matrix (regarded as a linear map) into cyclic $\mathbb{Z}$ modules (abelian groups). I immediately found the one dimensional kernel, but I do not know how to deal with the cokernel. It should have dimension one as well, but it might be a direct sum of cyclic groups. Since this question appeared in all kinds of prelim tests I feel these must be a universal way to solve it.

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The universal way to solve this kind of problem is provided by the Smith normal form.