Part BCSIR NET December 2024the-minimal-polynomial-divides-the-gcd

The minimal polynomial divides the gcd

Let $T : R^{4} \to R^{4}$ be an $R -$ linear transformation such that $(T^{2} + T + I) (T - 2 I) = 0$ and $(T^{2} + T + 2 I) (T^{2} - 4 I) = 0$ . Which of the following statements is FALSE?

A.T is diagonalizable over $R$ .
B.The characteristic polynomial of T is $(x - 2)^{4}$ .
C.The characteristic polynomial of T is $(x^{2} + x + 2) (x^{2} - 4)$ .✓
D.For every $R -$ linear transformation $S : R^{4} \to R^{4}$ , we have ST = TS.

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

The minimal polynomial divides the gcd

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