Principal ideal domain, $\forall x=(x_1,x_2)^t \in R^2~~\exists G \in SL_2(R) : Gx=(\gcd(x_1,x_2), 0)^t$

Question

Let $R$ be a principal ideal domain. Prove that for every $x=(x_1,x_2)^t \in R^2$ exists a matrix $G \in SL_2(R)$ for which $Gx=(\gcd(x_1,x_2), 0)^t$.

I think it's easy, but do not know how to start. Thanks.

Answer 1

Set $d=\gcd(x_1,x_2)$. Since $R$ is a PID there exist $a_1,a_2\in R$ such that $a_1x_1+a_2x_2=d$. Write $x_1=dy_1$, and $x_2=dy_2$. The matrix you are looking for is $$\left(\begin{array}{cc}a_1 &a_2\\-y_2 & y_1\end{array}\right).$$