Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 18 / b / ii / Solution

For each possibility in part (i), squaring the diagonal entries gives $M^2=I$. Equivalently, the linear transformation fixes a vector space basis after being applied twice. ThereforeGeometrically, reflecting twice in the same plane returns every vector to itself.