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 have

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 .

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