Question 137
For any positive integer $n$ and for any column matrix
$$
X = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} \in \mathcal{M}_{n,1}(\mathbb{Z}),
$$
we denote by $\delta(X)$ the greatest common divisor of the numbers $x_1, x_2, \dots, x_n$. Let $n \in \mathbb{N}$, $n \ge 2$, and $A \in \mathcal{M}_n(\mathbb{Z})$. Prove that the following statements are equivalent:
a) $|\det A| = 1$ and
b) $\delta(AX) = \delta(X)$, for any $X \in \mathcal{M}_{n,1}(\mathbb{Z})$.
$$
X = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} \in \mathcal{M}_{n,1}(\mathbb{Z}),
$$
we denote by $\delta(X)$ the greatest common divisor of the numbers $x_1, x_2, \dots, x_n$. Let $n \in \mathbb{N}$, $n \ge 2$, and $A \in \mathcal{M}_n(\mathbb{Z})$. Prove that the following statements are equivalent:
a) $|\det A| = 1$ and
b) $\delta(AX) = \delta(X)$, for any $X \in \mathcal{M}_{n,1}(\mathbb{Z})$.