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Part CCSIR NET June 2024minimal-and-characteristic-coincide-rarely

Minimal and characteristic coincide rarely

Let V (≠ {0}) be a finite dimensional vector space over R\mathbb{R} and T : V → V be a linear operator. Suppose that the kernel of T equals the image of T. Which of the following statements are necessarily true?

  1. A.The dimension of V is even✓ Correct
  2. B.The trace of T is zero✓ Correct
  3. C.The minimal polynomial of T cannot have two distinct roots✓ Correct
  4. D.The minimal polynomial of T is equal to its characteristic polynomial

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.

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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: Equivalent matrices are similar

More on this topic

The chapter behind this: Linear transformations, matrices and similarity — free to read

From Vector Spaces and Linear Maps › Linear transformations, matrix representation, change of basis

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