Which of the following statements is/are TRUE for every real orthogonal matrix $Q$ of order $n$?
Options:
(A) $Q^T Q = Q Q^T = I$
(B) $|\det(Q)| = 1$
(C) The inverse of $Q$ is equal to its transpose ($Q^{-1} = Q^T$)
(D) All eigenvalues of $Q$ have an absolute magnitude of $1$
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
For any real orthogonal matrix $Q$:
1. By definition, $Q^T Q = I \implies Q^{-1} = Q^T$ (A and C are true).
2. Taking determinants: $\det(Q^T Q) = [\det(Q)]^2 = 1 \implies \det(Q) = \pm 1$, so $|\det(Q)| = 1$ (B is true).
3. If $\lambda$ is an eigenvalue with eigenvector $x$: $Qx = \lambda x \implies x^H Q^T Q x = |\lambda|^2 x^H x$. Since $Q^T Q = I$, $|\lambda|^2 = 1 \implies |\lambda| = 1$ (D is true).