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Part BCSIR NET December 2025matching-power-sums-pins-the-characteristic-polynomial-not-the-matrix

Matching power sums pins the characteristic polynomial not the matrix

Let M,N∈M3(C)M, N \in M_{3}(\mathbb{C}) be such that Trace(Mᵏ) = Trace(Nᵏ), 1 ≤ k ≤ 3. Which of the following statements is necessarily true?

  1. A.M2=N2M^{2} = N^{2}
  2. B.M3=N3M^{3} = N^{3}
  3. C.The characteristic polynomials of M and N are the same.✓
  4. D.The minimal polynomials of M and N are the same.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

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50 are analysed free — try those first.

Related counterexample: Same characteristic polynomial ⇒ similar

More on this topic

The chapter behind this: Characteristic vs minimal polynomial — what each tells you — free to read

From Eigenvalues and Canonical Forms › Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

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