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

A.T has at least four linearly independent eigenvectors.✓
B.dim W ≥ 2.
C. $ker ((T - 2 I)^{2025}) = ker ((T - 2 I)^{2026})$ ✓
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

The data caps block sizes but does not pin the multiplicities

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