The problem seems not so hard. My confusion rise from the statement in the solution above that "This question is equivalent to reducing the matrix via row and column operations".
Please see the above picture.
I understand row-operation is valid, since it's just changing the base of $M$. However, I cannot understand why we can also take column-operations.
Please help!
You are actually allowed column operations in this situation, because $\mathbb{Z}^{3}$ admits an automorphism permuting the three copies of $\mathbb{Z}$; thus $\mathbb{Z}^{3}/<(13,9,2), (29,21,5), (2,2,2)> \simeq \mathbb{Z}^{3}/<(9,13,2),(21,29,5), (2,2,2)>$ and so on.
You normally don't want to use column operations on a matrix, because it is equivalent to renaming your "variables" which can be confusing. But since the "variables" correspond to the 3 copies of $\mathbb{Z}$ which all play the same role, there is no problem here.