If the characteristic roots of are λ₁ and λ₂, the characteristic roots of are:
2019
If the characteristic roots of are λ₁ and λ₂, the characteristic roots of are:
- A.
1/λ₁ , 1/λ₂
- B.
λ₁ + λ₂ , λ₁ − λ₂
- C.
λ₁ + λ₂ , |λ₁ − λ₂|
- D.
2λ₁ , 2λ₂
Attempted by 3 students.
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Correct answer: A
Key fact: If A is invertible and v is an eigenvector of A with eigenvalue λ (so Av = λv), then A⁻¹v = (1/λ)v.
Start from Av = λv for an eigenvector v and eigenvalue λ.
Apply A⁻¹ to both sides: A⁻¹Av = A⁻¹(λv) which gives v = λA⁻¹v, so A⁻¹v = (1/λ)v.
Therefore every eigenvalue λ of A produces an eigenvalue 1/λ of A⁻¹. Thus the characteristic roots of A⁻¹ are 1/λ₁ and 1/λ₂, provided λ₁ and λ₂ are nonzero (so A is invertible).