Part CCSIR NET December 2024cube-roots-of-unity-are-distinct-but-not-real

Cube roots of unity are distinct but not real

Let A = [[0, 0, 1], [1, 0, 0], [0, 1, 0]] and B = [[0, 0, 1, 0], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]]. Which of the following statements are true?

A.Both A and B are diagonalizable over $\mathbb{R}$ .
B.A is diagonalizable over $\mathbb{C}$ but not over $\mathbb{R}$ .✓ Correct
C.Neither A nor B is diagonalizable over $\mathbb{R}$ , but both A and B are diagonalizable over $\mathbb{C}$ .✓ Correct
D.Neither A nor B is diagonalizable over $\mathbb{C}$ .

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 base field or ring.

See pricing

50 are analysed free — try those first.

The trap it tests

Base field or ring

The answer changes with the field or ring you are working over.

Drill statements like this

Related counterexample: Diagonalisable ⇒ invertible

Cube roots of unity are distinct but not real

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