Solution to Exercise 3.19.
The trouble with the world is that the stupid are cocksure and the intelligent are full of doubt..
—Bertrand Russell (1872–1970)

Exercise 3.19. Prove that every affine transformation of the Euclidean plane has an inverse. (Hint. Write the inverse by using the adjoint. Refer to a linear algebra text.) 

Proof. Let A be the matrix of an affine transformation of the Euclidean plane.

 

By definition of an affine transformation, the determinant of A is nonzero. Hence, the inverse of A exists. From linear algebra results,

 

Thus,

 

Hence, the inverse of matrix A is a matrix of an affine transformation of the Euclidean plane.//