Solution to Exercise 3.44.
Behind the wall, the gods play; they play with numbers, of which the universe is made.
—Le Corbusier (1887-1965)

Exercise 3.44. Prove the inverse of an indirect isometry is an indirect isometry.

Proof. Let  be the matrix of an affine indirect isometry of the Euclidean plane. By the Corollary to Proposition 3.7, det(A) = –1. Hence,

 

Thus, A^–1 is the matrix of an affine indirect isometry of the Euclidean plane. Hence, the inverse of an affine indirect isometry of the Euclidean plane is an affine indirect isometry of the Euclidean plane.//