The reader should first try the following exercise, which motivates the later work.
Exercise: Prove that the set of real matrices
forms a group under the usual matrix multiplication.
This exercise comes with a guarantee that it will provoke at least a doubletake when first encountered. The usual reaction is to protest that these matrices are obviously singular, or to complain that the identity is missing from the set. Of course those familiar with this example can feel suitably smug in their rapid dismissal of these criticisms, but what was your reaction the first time you saw it?