Unitary matrices are invertible

Question

By definition, an unitary matrix $U$ is defined as $U U^H = I$.

How can we prove that an unitary matrix $U$ is invertible? It can happen that $U^H U \neq I$.

Answer 1

Unitary matrix is by definition a square matrix $U$ over $\mathbb C$ such that $$U\overline{U}^t = \overline{U}^tU = I. $$

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I'm assuming $U^H$ means the transpose of the conjugate. Using definition of unitarity and some linear algebra magic, we get $$\det (UU^H) = \det (U) \det (U^H) = \det (U)\det (\overline{U}) = \det (U)\overline{\det(U)} = 1, $$ i.e $|\det (U)|^2 = 1$, which implies $\det (U) = e^{i\varphi}$, hence $U$ is invertible.

Also, for square matrices, $AB = I$ implies $$BAB = B \Rightarrow BA = I. $$