Part CCSIR NET December 2024powers-stop-growing-at-the-minimal-polynomial

Powers stop growing at the minimal polynomial

For each n > 1, let V denote the $C -$ vector space of all n × n complex matrices and A ∈ V. Which of the following statements are necessarily true?

A.The set { $I, A, \dots, A^{n}$ } is linearly independent but the set { $I, A, \dots, A^{(} n^{2})$ } is not linearly independent.
B.If A is a singular matrix, then the set {I, A, …, A^k} spans a (k + 1)-dimensional subspace of V for all k ≤ rank(A).
C.The sets { $I, A, \dots, A^{n}$ } and { $I, A, \dots, A^{(} n^{2})$ } both span the same subspace of V.✓
D.The set { $I, A, \dots, A^{n}$ } is linearly dependent.✓

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.

50 are analysed free — try those first.

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