Should a Pauli $X$ matrix equal the identity matrix to be unitary?

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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?

The definition of unitary matrix is that its conjugate transpose should equal its inverse, i.e., $$U U^\dagger = U^\dagger U = I$$. You can check that this is indeed the case for the Pauli X matrix.
3The product of $U$ 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