Prove that the set of all 3 × 3 matrices with real entries of the form is a gr...

Prove that the set of all 3 × 3 matrices with real entries of the form

is a group. (Multiplication is defined by

This group, sometimes called the Heisenberg group after the Nobel Prize–winning physicist Werner Heisenberg, is intimately related to the Heisenberg Uncertainty Principle of quantum physics.)