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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?

  1. A.Both A and B are diagonalizable over .
  2. B.A is diagonalizable over but not over .✓
  3. C.Neither A nor B is diagonalizable over , but both A and B are diagonalizable over .✓
  4. D.Neither A nor B is diagonalizable over .

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: Diagonalisable ⇒ invertible

More on this topic

The chapter behind this: Diagonalisability — the criteria card — free to read

From Eigenvalues and Canonical Forms › Diagonalisability criteria

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