If the characteristic polynomial of a 3 × 3 matrix M over R (the set of real numbers) is λ³ – 5λ² + aλ + 24, where a ∈ R, and one eigenvalue of M is 3, then the smallest and largest eigenvalues of M are __________ .
(A) -4 and -2 respectively.
(B) -4 and 2 respectively.
(C) -2 and 4 respectively.
(D) 2 and 4 respectively.


Answer: (C)

Explanation: f(λ) = λ³ – 5λ² + aλ + 24

Now 3 is one of the roots of this equation. So,

33 - 5 x 32 + a x 3 + 24 = 0
27 - 45 + 3a + 24 = 0
3a + 6 = 0
a = -2 

So, the equation is,

λ3 – 5λ2 + -2λ + 24 

Now, by polynomial division we get,

(λ3 – 5λ2 + -2λ) + 24 / (λ - 3) = λ2 - 2λ-8 

Now find roots of,

λ2 - 2λ - 8 = 0
λ2 - 4λ + 2λ - 8 = 0
λ(λ - 4 ) + 2(λ - 4) = 0
(λ - 4)(λ + 2) = 0
λ = 4 , -2 

So, the eigenvalues are -2, 3 and 4. the smallest and largest eigenvalues of M are -2 and 4 respectively.

Option (C) is correct.

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