Exercise 21, § 4

Let C be a principal ideal domain, and let $A = (\alpha_{ij})$ be a square matrix of order n over C. a) Show that there exists a square matrix U of order n over C, with determinant 1, such that the last column of the matrix UA has its first $(n-1)$ entries zero, and its last entry equal to the gcd of the entries in the n-th column of $A$ (use Ex. 20, b)). b) Deduce from a) that there exists a square matrix V of order n over C, with determinant 1, such that the matrix VA = $(\beta_{ij})$ is lower triangular ("Hermite reduced form of A"). c) If W is a square matrix of order n over C, with determinant 1, such that WA = $(\gamma_{ij})$ is lower triangular, show that $\beta_{ii}$ and $\gamma_{ii}$ are associate for $1 \leq i \leq n$.