<p>Let \(A = \begin{bmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{bmatrix}\). Which of the following is true?</p>
Step-by-Step Solution
Key Concept: Recognize that A is an orthogonal matrix (A·Aᵀ = I) with determinant -1, making it a reflection matrix. Compute eigenvalues by solving det(A - λI) = 0 to identify its spectral properties.
<p><strong>Step 1: Verify orthogonality</strong></p><p>Compute Aᵀ = A (since A is symmetric)</p><p>AAᵀ = A² = <span style='font-family:monospace'>[[1,0,0],[0,1,0],[0,0,1]]</span> = I ✓</p><p><strong>Step 2: Find determinant</strong></p><p>det(A) = 0(−1·0 − 0·0) − 0 − (−1)(0·0 − (−1)·(−1))</p><p>det(A) = −1(0 − 1) = −1</p><p><strong>Step 3: Find eigenvalues</strong></p><p>Characteristic equation: det(A − λI) = −(λ+1)²(λ−1) = 0</p><p>Eigenvalues: λ = −1 (multiplicity 2), λ = 1 (multiplicity 1)</p><p><strong>Step 4: Key properties</strong></p><p>• A is orthogonal (det(A) = ±1, columns/rows orthonormal)</p><p>• det(A) = −1 indicates A is a reflection (not pure rotation)</p><p>• All eigenvalues have absolute value 1</p><p>• A³ = −A (can be verified)</p><p>∴ Answer: B</p>
Correct Answer: B