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Unitary matrices with rational entries

Source: Brazilian Undergrad MO 2022 Problem 1

May 12, 2022

Brazilian Undergrad MOMatrix algebraSquare matrixlinear algebraBrazilian Undergrad MO 2021

Problem Statement

Consider the matrices like

M= \left( \begin{array}{ccc} a & b & c \\ c & a & b \\ b & c & a \end{array} \right)

such that

det(M) = 1

.
Show that
a) There are infinitely many matrices like above with

a,b,c \in \mathbb{Q}

b) There are finitely many matrices like above with

a,b,c \in \mathbb{Z}