# Unitary matrices are invertible.

Put content here**Fact.** Every unitary matrix is invertible.
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By definition, if $U$ is unitary then $U^* U = I$, which means $U^{-1} = U^*$. The inverse of a unitary matrix is its conjugate transpose, which always exists.
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Furthermore:
- $|\det(U)| = 1$ (the determinant has absolute value 1)
- The eigenvalues of a unitary matrix all have absolute value 1
- The product of unitary matrices is unitary
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**Example.** For $U = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, we have $U^{-1} = U^* = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.

# Parents

* Unitary matrices