completing a primitive integer vector into an integer matrix of determinant 1

Question

This is probably well known in algebraic number theory, in particular Minkowski lattice theory, but I am given an integer vector of dimension $n$ whose components are relatively prime, meaning there is an integer linear combination of them that equals 1. Is it true that I can always find $n-1$ other integer vectors such that they form an element of $SL(n, \mathbb{Z})$? I tried to think in terms of cofactor expansion of determinant, but that was only useful for $n=2$. Thinking geometrically also bore no fruit. Thanks!

Answer 1

HINT: This is a particular case of Smith normal form theorem.

Answer 2

You can find the construction of such a basis in M. Newman's book "Integral Matrices". It's Theorem II.$1$ on Page 13.