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Part BCSIR NET December 2024orthogonally-diagonalizable-means-symmetric

Orthogonally diagonalizable means symmetric

For any matrix P, the transpose of P is denoted by Pᵗ. Consider the real matrix A = [[1, 1, 0], [1, 2, 1], [1, 1, 2]]. Which of the following statements is true?

  1. A.There exists a real invertible matrix P such that PAP⁻1^{1} is a diagonal matrix and PᵗP=I3P = I_{3}.
  2. B.There exists a real invertible matrix P such that PAP⁻1^{1} is a diagonal matrix and PᵗP≠I3P \ne I_{3}.✓ Correct
  3. C.One of the eigenvalues of A is not real.
  4. D.A has only real eigenvalues and it is not diagonalizable over R\mathbb{R}.

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 converse assumed.

See pricing

50 are analysed free — try those first.

The trap it tests

Converse assumed

The theorem runs one way. You used it in the other.

Drill statements like this

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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