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 $I_{2}$ denote the 2 × 2 identity matrix. Which of the following statements are necessarily true?

A.If A and B are upper triangular, then M is diagonalizable over $R$ .
B.If A and B are diagonalizable over $R$ , then M is diagonalizable over $R$ .
C.If A and B are diagonalizable over $R$ , then there exists $λ \in R$ such that $M = λ I_{2}$ .
D.There exists $λ \in R$ such that $M^{2} = λ I_{2}$ .✓

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.

See pricing

50 are analysed free — try those first.

Related counterexample: AB = I ⇒ BA = I for all matrices

A two by two commutator squares to a scalar

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