Matrices & Determinants
Eigenvalues of a Matrix
Grade 12
Question:
<p>Which of the following matrices have eigen values as 1 and \(-1\)?</p>
<p>\(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)</p>
Step-by-Step Solution
Key Concept: A matrix has eigenvalues 1 and -1 if and only if det(A) = (1)(-1) = -1 and the characteristic polynomial factors as (λ-1)(λ+1). Use the trace and determinant: trace(A) = 1 + (-1) = 0.
<p><strong>Step 1:</strong> For a 2×2 matrix with eigenvalues 1 and -1, use the properties:</p><p>• Product of eigenvalues = det(A) = 1 × (-1) = <strong>-1</strong></p><p>• Sum of eigenvalues = trace(A) = 1 + (-1) = <strong>0</strong></p><p><strong>Step 2:</strong> Check each option for:</p><p>✓ trace(A) = 0 (diagonal elements sum to 0)</p><p>✓ det(A) = -1</p><p><strong>Step 3:</strong> Verify by computing characteristic polynomial det(A - λI) = 0:</p><p>For correct option: (λ - 1)(λ + 1) = λ² - 1 = 0 ⟹ λ = ±1</p><p>∴ Answer: A</p>
Correct Answer: A