Part CCSIR NET June 2024minimal-polynomial-gives-the-eigenvalues

Minimal polynomial gives the eigenvalues

Let $T : R^{4} \to R^{4}$ be a linear map with four distinct eigenvalues and satisfying $T^{4} - 15 T^{2} + 10 T + 24 I = 0$ . Which of the following statements are necessarily true?

A.There exists a non-zero vector $v_{1} \in R^{4}$ such that Tv $_{1} = 2 v_{1}$ ✓
B.There exists a non-zero vector $v_{2} \in R^{4}$ such that Tv $_{2} = v_{2}$
C.For every non-zero vector $v \in R^{4}$ , the set {2v, 3Tv} is linearly independent
D.T is a one-one function✓

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

Minimal polynomial gives the eigenvalues

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