If , , then is
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: A
Solution
Find the characteristic polynomial of A using det(A - λI) = 0.
det(A - λI) = (1-λ)[(1-λ)(4-λ) + 2] = (1-λ)[λ² - 5λ + 6] = (1-λ)(λ-2)(λ-3)
So eigenvalues are 1, 2, 3, and the characteristic polynomial is:
(λ-1)(λ-2)(λ-3) = λ³ - 6λ² + 11λ - 6 = 0
By Cayley-Hamilton: A³ - 6A² + 11A - 6I = 0
Dividing both sides by A (multiplying by A⁻¹):
A² - 6A + 11I - 6A⁻¹ = 0 → 6A⁻¹ = A² - 6A + 11I
Comparing with 6A⁻¹ = A² + cA + dI gives c = -6 and d = 11, so (c, d) = (-6, 11).
det(A - λI) = (1-λ)[(1-λ)(4-λ) + 2] = (1-λ)[λ² - 5λ + 6] = (1-λ)(λ-2)(λ-3)
So eigenvalues are 1, 2, 3, and the characteristic polynomial is:
(λ-1)(λ-2)(λ-3) = λ³ - 6λ² + 11λ - 6 = 0
By Cayley-Hamilton: A³ - 6A² + 11A - 6I = 0
Dividing both sides by A (multiplying by A⁻¹):
A² - 6A + 11I - 6A⁻¹ = 0 → 6A⁻¹ = A² - 6A + 11I
Comparing with 6A⁻¹ = A² + cA + dI gives c = -6 and d = 11, so (c, d) = (-6, 11).
[AIEEE 2005]