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Part CCSIR NET December 2024over-the-reals-an-irreducible-quadratic-forces-even-degree

Over the reals an irreducible quadratic forces even degree

Which of the following statements are true?

  1. A.There exists a 5 × 5 real matrix whose minimal polynomial is (x2+x+1)x3(x^{2} + x + 1)x^{3}.✓ Correct
  2. B.There exists a 5 × 5 real matrix whose minimal polynomial is (x2+x+1)2(x^{2} + x + 1)^{2}.
  3. C.There exists a 5 × 5 complex matrix whose minimal polynomial is (x2+x+1)2x(x^{2} + x + 1)^{2}x.✓ Correct
  4. D.There exists a 5 × 5 complex matrix whose minimal polynomial is (x2+x+1)2(x^{2} + x + 1)^{2}.✓ Correct

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, and why this one tests base field or ring.

See pricing

50 are analysed free — try those first.

The trap it tests

Base field or ring

The answer changes with the field or ring you are working over.

Drill statements like this

Related counterexample: Every real matrix has a Jordan form over ℝ

More on this topic

The chapter behind this: Jordan canonical form — free to read

From Eigenvalues and Canonical Forms › Jordan canonical form

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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