# How do the elementary divisors of a matrix over a PID change if it is enlarged by a vector?

Given an $$n\times m$$ matrix $$A$$ over a PID $$R$$, there exist $$0\leq r\leq \min\{m,n\}$$ and $$e_1, \dotsc, e_r \in R\setminus\{0\}$$ with $$e_1\mid e_2\mid \cdots \mid e_r$$ as well as matrices $$U,V\in \textrm{Gl}_n(R)$$ s.t. $$UAV = \begin{pmatrix} e_1 & & & | & \\ & \ddots & & |&\\ &&e_r & | & \\ \hline & & & | & 0&\\ \end{pmatrix}.$$ (This is the Smith normal form, with the $$e_i$$ being the elementary divisors of $$A$$.)

I'm interested in the change of the elementary divisors if we increase the size of our given matrix by a vector (i.e. $$A \leftrightarrow (A\mid v)$$ for some $$v\in R^n$$).

We know that the product of the first $$l$$ elementary divisors can not increase (w.r.t. divisibility), thus the amount of elementary divisors can not decrease. (Because $$e_1 \cdots e_l$$ is the greatest common divisor of all $$l\times l$$ minors.)

I would like to apply this to calculate the distribution of finite abelian $$p$$-groups as cokernels of $$p$$-adic matrices of size $$n \times (n+1)$$. The case for $$n\times n$$ matrices was already calculated by Friedman & Washington in On the distribution of divisor class groups of curves over a finite field. Their distribution converges to the Cohen-Lenstra distribution for $$n\rightarrow \infty$$, thus this would give me the distribution of finite abelian $$p$$-groups given by dividing a randomly (w.r.t. the Cohen-Lenstra distribution) drawn $$p$$-group by the subgroup generated by a random (w.r.t. the uniform distribution) element.