# GATE | Sudo GATE 2020 Mock I (27 December 2019) | Question 30

If the characteristic polynomial of a 3 × 3 matrix M over R (the set of real numbers) is λ^{3} – 5λ^{2} + 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) = λ^{3} – 5λ^{2} + aλ + 24

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

3^{3}- 5 x 3^{2}+ a x 3 + 24 = 0 27 - 45 + 3a + 24 = 0 3a + 6 = 0 a = -2

So, the equation is,

λ^{3}– 5λ^{2}+ -2λ + 24

Now, by polynomial division we get,

(λ^{3}– 5λ^{2}+ -2λ) + 24 / (λ - 3) = λ^{2}- 2λ-8

Now find roots of,

λ^{2}- 2λ - 8 = 0 λ^{2}- 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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