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Eigenvalues and eigenvectors [2,−2,2;1,1,1;…]

Short answer

The characteristic equation λ³ − 2λ² − 4λ + 8 = 0 yields eigenvalues 2, 2, −2. Determinant = product = −8. Eigenvalues of A⁻¹ are the reciprocals: 1/2, 1/2, −1/2.

Find the eigenvalues of the matrix A = [2 −2 2; 1 1 1; 1 3 −1]. Hence use them to compute a) |A| b) the eigenvalues of A⁻¹.

A = [2 −2 2; 1 1 1; 1 3 −1]

Eigenvalues

  • Trace = 2
  • Sum of principal minors = (−1 − 3) + (−2 − 2) + (2 + 2) = −4
  • |A| = 2(−1 − 3) + 2(−1 − 1) + 2(3 − 1) = −8
λ³ − 2λ² − 4λ + 8 = 0  ⇒  (λ − 2)²(λ + 2) = 0

Eigenvalues: 2, 2, −2.

(a) Determinant

The determinant equals the product of the eigenvalues:

|A| = 2 × 2 × (−2) = −8

(Direct expansion also gives −8, which confirms the result.)

(b) Eigenvalues of A⁻¹

If λ is an eigenvalue of a non-singular matrix A, then 1/λ is an eigenvalue of A⁻¹ (from AX = λX ⇒ X = λA⁻¹X ⇒ A⁻¹X = (1/λ)X). Since |A| ≠ 0, A⁻¹ exists:

Eigenvalues of A⁻¹: 1/2, 1/2, −1/2.

✓ Answer checked by Claude Opus 5.5 (high effort)

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