Let A be a non-zero 3 × 3 matrix with integer entries. Let λi ∈ &mathbb;C, 1 ≤ i ≤ 3 be all the eigenvalues of A (not necessarily distinct). Which of the following statements is necessarily true?

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Mathematics · Linear Algebra

Verified Solution4 OptionsUpdated: October 2026👁️1 View

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Correct answer

There exists a cubic polynomial f(X) ∈ &mathbb;Q[X] such that f(λi) = 0 for all 1 ≤ i ≤ 3.

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Answer 1

If f(X) ∈ &mathbb;Q[X] is a degree 2 polynomial such that f(λi) = 0 for all 1 ≤ i ≤ 3, then f(A) = 0.

Answer 2

If f(X) ∈ &mathbb;Q[X] is such that f(λi) = 0 for all 1 ≤ i ≤ 3, then f(A) = 0.

Answer 3

There exists a quadratic polynomial f(X) ∈ &mathbb;Q[X] such that f(λi) = 0 for all 1 ≤ i ≤ 3.

Answer 4

There exists a cubic polynomial f(X) ∈ &mathbb;Q[X] such that f(λi) = 0 for all 1 ≤ i ≤ 3.

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