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Romanian National Olympiad 2018 - Grade 11 - problem 1

Source: Romania NMO - 2018

April 7, 2018
greatest common divisorvectorlinear algebramatrix

Problem Statement

Let n≥2n \geq 2 be a positive integer and, for all vectors with integer entries X=(x1x2⋮xn)X=\begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} let δ(X)≥0\delta(X) \geq 0 be the greatest common divisor of x1,x2,…,xn.x_1,x_2, \dots, x_n. Also, consider A∈Mn(Z).A \in \mathcal{M}_n(\mathbb{Z}). Prove that the following statements are equivalent: <spanclass=′latex−bold′>i)</span><span class='latex-bold'>i) </span> ∣det⁡A∣=1|\det A | = 1 <spanclass=′latex−bold′>ii)</span><span class='latex-bold'>ii) </span> δ(AX)=δ(X),\delta(AX)=\delta(X), for all vectors X∈Mn,1(Z).X \in \mathcal{M}_{n,1}(\mathbb{Z}).
Romeo Raicu
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