Question 148
For any nonzero natural number $n$ and any column matrix
$$
\mathbf{X} = \left(\begin{array}{c}
x_{1} \\
x_{2} \\
\vdots \\
x_{n}
\end{array}\right) \in \mathcal{M}_{n, 1}(\mathbb{Z})
$$
we denote by $\delta(\mathbf{X})$ the greatest common divisor of the numbers $x_{1}, x_{2}, \ldots, x_{n}$. Let $n \in \mathbb{N}, n \geq 2$, and $\mathbf{A} \in \mathcal{M}_{n}(\mathbb{Z})$. Show that the following two statements are equivalent:
(a) $|\operatorname{det} \mathbf{A}|=1$ and
(b) $\delta(\mathbf{A X})=\delta(\mathbf{X})$, for every $\mathbf{X} \in \mathcal{M}_{n, 1}(\mathbb{Z})$.
(The greatest common divisor of some integers is a natural number.)
$$
\mathbf{X} = \left(\begin{array}{c}
x_{1} \\
x_{2} \\
\vdots \\
x_{n}
\end{array}\right) \in \mathcal{M}_{n, 1}(\mathbb{Z})
$$
we denote by $\delta(\mathbf{X})$ the greatest common divisor of the numbers $x_{1}, x_{2}, \ldots, x_{n}$. Let $n \in \mathbb{N}, n \geq 2$, and $\mathbf{A} \in \mathcal{M}_{n}(\mathbb{Z})$. Show that the following two statements are equivalent:
(a) $|\operatorname{det} \mathbf{A}|=1$ and
(b) $\delta(\mathbf{A X})=\delta(\mathbf{X})$, for every $\mathbf{X} \in \mathcal{M}_{n, 1}(\mathbb{Z})$.
(The greatest common divisor of some integers is a natural number.)