This is one in a series of posts working through the exercises in the Quantum Country online introduction to quantum computing and related topics. The exercises in the original document are not numbered; I have added my own numbers for convenience in referring to them.
Exercise 13. Show that the product of two unitary matrices and is itself a unitary matrix.
Answer: Since and are unitary matrices we have and .
Now consider the product . We have
Since the product matrix is unitary. Thus the product of two unitary matrices and is itself a unitary matrix.