Which of the following statements are TRUE for any real orthogonal matrix $Q$ ($Q^T Q = I$)?
Options:
(A) The determinant of $Q$ can only be $+1$ or $-1$
(B) The absolute value (modulus) of each of its eigenvalues is equal to $1$
(C) The column vectors of $Q$ form an orthonormal set
(D) The matrix $Q$ is always symmetric ($Q = Q^T$)
Answer Key:
A, B, C
Step-by-Step Solution & Explanation
- A is true: $\det(Q^T Q) = [\det(Q)]^2 = 1 \implies \det(Q) = \pm 1$.
- B is true: For $Q x = \lambda x$, $\|Q x\| = \|x\| \implies |\lambda| = 1$.
- C is true: $Q^T Q = I$ means $q_i^T q_j = \delta_{ij}$, an orthonormal basis.
- D is false: Orthogonal matrices need not be symmetric (e.g., standard 2D rotation matrix for $\theta \ne 0$).