Solution to Exercise 3.42.
If I were asked to enumerate ten educational stupidities, the giving of grades would head the list… If I can't give a child a better reason for studying than a grade on a report card, I ought to lock my desk and go home and stay there.
—Dorothy De Zouche

Exercise 3.42. Prove Proposition 3.10.
Proposition 3.10. The set of affine isometries of the Euclidean plane is a group.

 

Proof. The associative property follows from matrix multiplication and the identity follows from closure and the inverse property; therefore, we only need show closure and the inverse property. By Proposition 3.8, the set of affine isometries is closed under matrix multiplication. By Proposition 3.9, the inverse of an affine direct isometry is an affine isometry. Hence, we only need to show the inverse of an affine indirect isometry is an affine isometry. 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 isometry of the Euclidean plane is an affine isometry of the Euclidean plane. Therefore, the set of affine isometries of the Euclidean plane is a group under matrix multiplication.//