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Part CCSIR NET June 2025two-nilpotent-matrices-can-multiply-to-something-with-eigenvalue-1

Two nilpotent matrices can multiply to something with eigenvalue 1

Let A, B be distinct 2 × 2 real matrices. Which of the following statements are true?

  1. A.If A is invertible, then AB and BA have the same minimal polynomial.✓
  2. B.If 0 is an eigenvalue of A, then 0 is an eigenvalue of AB.✓
  3. C.If 0 is the only eigenvalue of A and of B, then 0 is the only eigenvalue of AB.
  4. D.If AB and BA have the same minimal polynomial, then either A or B is invertible.

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 standard counterexample.

See pricing

50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

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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