MathDB

Problem 1

Part of 2021 Brazil Undergrad MO

Problems(1)

Unitary matrices with rational entries

Source: Brazilian Undergrad MO 2022 Problem 1

5/12/2022
Consider the matrices like
M=(abccabbca)M= \left( \begin{array}{ccc} a & b & c \\ c & a & b \\ b & c & a \end{array} \right)
such that det(M)=1det(M) = 1.
Show that
a) There are infinitely many matrices like above with a,b,c∈Qa,b,c \in \mathbb{Q}
b) There are finitely many matrices like above with a,b,c∈Za,b,c \in \mathbb{Z}
Brazilian Undergrad MOMatrix algebraSquare matrixlinear algebraBrazilian Undergrad MO 2021