Problem

Source: Romania NMO - 2018

Tags: greatest common divisor, vector, linear algebra, matrix



Let $n \geq 2$ be a positive integer and, for all vectors with integer entries $$X=\begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}$$let $\delta(X) \geq 0$ be the greatest common divisor of $x_1,x_2, \dots, x_n.$ Also, consider $A \in \mathcal{M}_n(\mathbb{Z}).$ Prove that the following statements are equivalent: $\textbf{i) }$ $|\det A | = 1$ $\textbf{ii) }$ $\delta(AX)=\delta(X),$ for all vectors $X \in \mathcal{M}_{n,1}(\mathbb{Z}).$ Romeo Raicu