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 $x^{2} + x + 1$ and let $B = A + I_{4}$ . Which of the following statements are necessarily true?

A.The minimal polynomial of B is $x^{2} + x + 1$
B.The minimal polynomial of B is $x^{2} - x + 1$ ✓
C. $B^{3} = I_{4}$
D. $B^{3} + I_{4} = 0$ ✓

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.

See pricing

50 are analysed free — try those first.

Related counterexample: Same characteristic polynomial ⇒ similar

Shifting shifts the minimal polynomial

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