Part CCSIR NET December 2024the-commutator-is-the-derivative-itself

The commutator is the derivative itself

For a variable x, consider the real vector space V = {ax $^{3} +$ bx $^{2} +$ cx + d | $a, b, c, d \in R$ }. Let D : V → V be the linear transformation where D(f) is the derivative of f with respect to x, and M : V → V be the linear transformation M(f) = xD(f). Which of the following statements are true?

A.DM ≠ MD.✓
B.D + M is invertible.
C.DM is invertible.
D.rank(DM) = rank(MD).

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.

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50 are analysed free — try those first.

Related counterexample: Equivalent matrices are similar

The commutator is the derivative itself

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