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Part CCSIR NET June 2024shifting-shifts-the-minimal-polynomial

Shifting shifts the minimal polynomial

Let A be a 4 × 4 real matrix whose minimal polynomial is x2+x+1x^{2} + x + 1 and let B=A+I4B = A + I_{4}. Which of the following statements are necessarily true?

  1. A.The minimal polynomial of B is x2+x+1x^{2} + x + 1
  2. B.The minimal polynomial of B is x2−x+1x^{2} - x + 1✓ Correct
  3. C.B3=I4B^{3} = I_{4}
  4. D.B3+I4=0B^{3} + I_{4} = 0✓ 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 execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

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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