Show that any orthogonal matrix has determinant 1 or -1

Question

Hello fellow users of this forum:

Show that for any orthogonal matrix Q, either det(Q)=1 or -1.

Thanks

Answer 1

Not sure what's wrong with using the transpose, but here it goes.

Since $Q$ is orthogonal, $QQ^T = I = Q^TQ$ by definition. Using the fact that $\det(AB) = \det(A) \det(B)$, we have $\det(I) = 1 = \det(QQ^T) = \det(Q) \det(Q^T) = \det(Q) \det(Q) = [\det(Q)]^2$.

Since we have $[\det(Q)]^2 = 1$, then $\det(Q) = \pm \sqrt{1} = \pm 1$.