What's the trick for proving one eigenvalue of orthogonal matrix is $-1$ if the determinant is $-1$?

Question

Obviously, the magnitude of the orthogonal matrix is 1, which is easy to prove.. However, I wonder how can one prove that the eigenvalue of an orthogonal matrix is $-1$, if the determinant of this matrix is $-1$?

Thanks a lot.

Thanks for the help that I can prove it now from the conjugate pairs point of view.

Is there a method by proving that $\det(I+A)=0$?

Answer 1

Hint: nonreal eigenvalues of real orthogonal matrices must occur in conjugate pairs, and the product of a conjugate pair of complex number is ...