I noticed the following description here:

Details

The result is given in the form {u,r,v}, where u and v are unimodular matrices, r is a diagonal matrix with each diagonal entry dividing the next one, and u.m.v==r.

The unimodular matrices u and v are integer matrices with Abs[Det[u]]==1, and their inverses are also integer matrices.

As you can see, there are two unimodular integer matrices, namely, u and v. Therefore, I think the above description should be strengthened as follows:

The unimodular matrices u and v are integer matrices with Abs[Det[u]]==1 and Abs[Det[v]]==1, and their inverses are also integer matrices.

Regards,
Zhao