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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−\mathbb{C}-vector space of all n × n complex matrices and A ∈ V. Which of the following statements are necessarily true?

  1. A.The set {I,A,…,AnI, A, \dots, A^{n}} is linearly independent but the set {I,A,…,A(n2)I, A, \dots, A^(n^{2})} is not linearly independent.
  2. 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).
  3. C.The sets {I,A,…,AnI, A, \dots, A^{n}} and {I,A,…,A(n2)I, A, \dots, A^(n^{2})} both span the same subspace of V.✓ Correct
  4. D.The set {I,A,…,AnI, A, \dots, A^{n}} is linearly dependent.✓ 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 finite-dimensional intuition.

See pricing

50 are analysed free — try those first.

The trap it tests

Finite-dimensional intuition

Something true in ℝⁿ, or in a nice space, assumed in general.

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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