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Part CCSIR NET December 2024a-two-by-two-commutator-squares-to-a-scalar

A two by two commutator squares to a scalar

Let A, B be 2 × 2 matrices with real entries, and M = AB − BA. Let I2I_{2} denote the 2 × 2 identity matrix. Which of the following statements are necessarily true?

  1. A.If A and B are upper triangular, then M is diagonalizable over R\mathbb{R}.
  2. B.If A and B are diagonalizable over R\mathbb{R}, then M is diagonalizable over R\mathbb{R}.
  3. C.If A and B are diagonalizable over R\mathbb{R}, then there exists λ∈R\lambda \in \mathbb{R} such that M=λI2M = \lambda{}I_{2}.
  4. D.There exists λ∈R\lambda \in \mathbb{R} such that M2=λI2M^{2} = \lambda{}I_{2}.✓ 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 standard counterexample.

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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: AB = I ⇒ BA = I for all matrices

More on this topic

The chapter behind this: Determinants, trace and block matrices — free to read

From Determinants and Matrix Tricks › Determinants, trace, block matrices, rank inequalities

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