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Primarily, here $A$ can be either real or complex matrix and as $A$ is orthogonal, hence all the eigenvalues of $A$ are of the form $e^{i\Delta}$ for $i^2=-1$ and real $\Delta$.

I tried to show it by induction but I couldn't prove for the $n=2$ case that $1$ is an eigenvalue of $A$. Can someone please provide me with a short hint?

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Every $n \times n$ orthogonal matrix $M$ with determinant $1$ defines an orientation-preserving rotation on the $n-1$ sphere $S^{n-1} \subset \mathbb{R}^n$, and this rotation has a fixed point at $x$ iff $x$ is an eigenvector of $M$ with eigenvalue $1$. The problem statement is thus a consequence of the well-known topological fact that every diffeomorphism from an even-dimensional sphere to itself of degree different to $-1$ has a fixed point.

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