If the characteristic roots of A = [[3, 7], [2, 5]] are λ₁ and λ₂, what are…
2019
If the characteristic roots of A = [[3, 7], [2, 5]] are λ₁ and λ₂, what are the characteristic roots of A⁻¹ = [[5, −7], [−2, 3]]?
Answer: A. 1/λ₁ , 1/λ₂ — ConceptFor an invertible matrix A, an eigenpair Av = λv with λ ≠ 0 becomes an eigenpair of A⁻¹ after applying the inverse: A⁻¹v = (1/λ)v. Equivalently,…
- A.
1/λ₁ , 1/λ₂
- B.
λ₁ + λ₂ , λ₁ − λ₂
- C.
λ₁ + λ₂ , |λ₁ − λ₂|
- D.
2λ₁ , 2λ₂
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Show answer & explanation
Correct answer: A
Concept
For an invertible matrix A, an eigenpair Av = λv with λ ≠ 0 becomes an eigenpair of A⁻¹ after applying the inverse: A⁻¹v = (1/λ)v.
Equivalently, inversion maps every eigenvalue λ to its reciprocal 1/λ while preserving the associated eigenvector.
Application — ordered working
For A = [[3, 7], [2, 5]], det(A) = 3·5 − 7·2 = 1, so A is invertible.
Using the 2 × 2 inverse formula, A⁻¹ = (1/det(A))[[5, −7], [−2, 3]] = [[5, −7], [−2, 3]], which is the second matrix in the stem.
If Av = λ₁v for an eigenvector v, multiply by A⁻¹ to obtain v = λ₁A⁻¹v, hence A⁻¹v = (1/λ₁)v. The same argument maps λ₂ to 1/λ₂.
Cross-check
The determinant equals the product of the eigenvalues. Here det(A) = det(A⁻¹) = 1, and (1/λ₁)(1/λ₂) = 1/(λ₁λ₂) = 1, so the reciprocal pair has the required product.
Result
The characteristic roots of A⁻¹ are 1/λ₁ and 1/λ₂.