I know that using an algorithm similar to Gauss' one, I can find the invariant factors of a given matrix (let's say $A=(a_{i,j})\in\mathcal M_n(\mathbb Z)$).
We first put in the top left corner $a_{1,1}$ the smallest value of the matrix $\min_{i,j}|a_{i,j}|$ and make 0s appear on its row and column. Then we repeat.
However I don't understand why when determining the matrix of the new base the operations are only applied on the rows and not also on the columns.