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Part CCSIR NET June 2025the-data-caps-block-sizes-but-does-not-pin-the-multiplicities

The data caps block sizes but does not pin the multiplicities

Let T:C7→C7T : \mathbb{C}^{7} \to \mathbb{C}^{7} be aC−a \mathbb{C}-linear operator with eigenvalues 2, 3 and 5. Consider the subspace W := {v∈C7:(T−5I)kv=0v \in \mathbb{C}^{7} : (T - 5I)^{k}v = 0 for some integer k > 0} of C7\mathbb{C}^{7}. Suppose that (T−2I)2(T−3I)2(T−5I)2=0(T - 2I)^{2}(T - 3I)^{2}(T - 5I)^{2} = 0. Which of the following statements are necessarily true?

  1. A.T has at least four linearly independent eigenvectors.✓ Correct
  2. B.dim W ≥ 2.
  3. C.ker⁡((T−2I)2025)=ker⁡((T−2I)2026)\ker((T - 2I)^{2025}) = \ker((T - 2I)^{2026})✓ Correct
  4. D.(T − 2I)(T − 3I) is a nilpotent operator.

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 invariants don't determine.

See pricing

50 are analysed free — try those first.

The trap it tests

Invariants don't determine

Two objects sharing an invariant were treated as the same object.

Drill statements like this

Related counterexample: Every real matrix has a Jordan form over ℝ

More on this topic

The chapter behind this: Jordan canonical form — free to read

From Eigenvalues and Canonical Forms › Jordan canonical form

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