Let $Q$ be a real $n \times n$ orthogonal matrix ($Q^T Q = I$). Which of the following mathematical properties must ALWAYS hold for $Q$?

(A) $\det(Q) = \pm 1$

(B) All eigenvalues of $Q$ have absolute magnitude equal to 1 ($|\lambda| = 1$)

(C) The inverse of $Q$ equals its transpose ($Q^{-1} = Q^T$)

(D) $Q$ must be a symmetric matrix ($Q = Q^T$)

Answer Key:

A, B, C

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Inversion Property:** By definition, $Q^T Q = I \implies Q^{-1} = Q^T$ (Statement C is TRUE). 2. **Stage 2: Determinant Property:** $\det(Q^T Q) = \det(Q^T)\det(Q) = [\det(Q)]^2 = \det(I) = 1 \implies \det(Q) = \pm 1$ (Statement A is TRUE). 3. **Stage 3: Eigenvalue Modulus:** For any eigenvector $x$ with eigenvalue $\lambda$: $|Qx|^2 = x^T Q^T Q x = x^T x = |x|^2$. But $|Qx|^2 = |\lambda|^2 |x|^2 \implies |\lambda| = 1$ (Statement B is TRUE). 4. **Stage 4: Counter-Example to Symmetry:** Consider standard 2D rotation $Q = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$. $Q$ is orthogonal, but $Q \ne Q^T$. Thus Statement D is FALSE. 5. **Stage 5: Official Reference:** Gilbert Strang, *Linear Algebra and Its Applications*, Chapter 5.

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