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My understanding is that any unitary matrix must have its inverse be equal to its conjugate transpose.

Looking at the pauli x gate as shown here: $$\begin{bmatrix}0&1\\1&0\end{bmatrix}$$

It is its own inverse which is equal, of course, to its own conjugate transpose.

However, isn't it also true that neither of these form an identity matrix? And isn't this a requirement for being considered unitary?

So in UU†=U†U=I, how does the =I make sense if the conjugate transpose does not have to equal the id matrix? sorry to be dense, appreciate your breaking it down here. – neutrino – 2019-09-09T19:35:42.337

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productof $U$ with its conjugate transpose $U^\dagger$ should equal the identity matrix. The conjugate transpose $U^\dagger$ does not need to equal $I$. Matrix multiplication is not, in general, commutative. – Mark S – 2019-09-09T21:08:29.797