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 7. Find a matrix other than , , or that is unitary.
Answer: Both and have two zero entries and two entries with value 1, such that the 1 values end up multiplying each other to produce a 1 value in the resulting matrix or .
This raises the possibility of having a matrix where the value ends up multiplying the value , since . Two possible candidate matrices with two entries with value are and .
We also have
Consider also the matrix . For it we have
Finally, for the matrix we have
So , , , and are all examples of unitary matrices other than , , or .