Exercise 1, § 4

Let U = (\alpha_{ij}) be a matrix with m rows and n columns over the principal ideal domain A. a) Suppose first of all that the a_i are setwise coprime. Show that there exist two invertible square matrices P and Q with entries in A, such that one of the entries of the matrix PUQ is equal to 1. (For each \alpha_{ij} \neq 0, let S(\alpha_{ij}) be the sum of the exponents in a decomposition of a_i into irreducible elements, and let s(U) be the least of the numbers s(\alpha_{ij}); if s(U) > 0, show that there exist two invertible square matrices R and S such that

use Bezout's identity, and restrict the matrices R and S considered, after permuting the rows and columns if necessary, to those of the form $\begin{pmatrix} T & 0 \\ 0 & I \end{pmatrix}$, where T is a 2 x 2 matrix and I an identity matrix, or to products of such matrices and permutation matrices.) b) If \delta_1 is a gcd of the entries of U, show that there exist two invertible matrices P_1, Q_1, such that

where all the entries of U_1 are divisible by \delta_1 (use a)). c) For A = \mathbf{Z}, use b) to obtain a method of calculating the invariant factors of an explicit matrix with entries in \mathbf{Z}. Apply this method to the matrix

EXERCISES