If an orthogonal matrix has determinant -1 then it has -1 as an eigenvalue

orthogonal-matrices

Solution 1:

The matrix is diagonalizable over $\mathbb C$, so the determinant is the product of the eigenvalues.

The complex eigenvalues that are not real come in conjugate pairs, and the product of two conjugate eigenvalues is a positive real. So there has to be at least one negative real eigenvalue.

The only negative real that can be an eigenvalue of an orthogonal matrix (which preserves the Euclidean norm of a vector) is $-1$.

Actually it's not necessary to appeal to diagonalizability; just considering the characteristic polynomial will do.

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