I have an algebra exam in a few weeks and, if the past papers are anything to go by, it seems likely that there will be a question on finding the Smith normal form of a 4x4 matrix with entries in $\mathbb{Z}$ using unimodular elementary row/column operations.
I understand the process but I find it very difficult to compute the SNF without some sort of arithmetic error, especially when the numbers can get large and not much time is available. Because of the nature of the SNF, most errors will still result in a reasonable looking matrix so it is hard to spot if I have made a mistake.
What methods could you I use to spot a mistake given the final SNF? In other words, what simple properties of a matrix are invariant under unimodular elementary row/column operations?