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?

A.There exists a real invertible matrix P such that PAP⁻ $^{1}$ is a diagonal matrix and Pᵗ $P = I_{3}$ .
B.There exists a real invertible matrix P such that PAP⁻ $^{1}$ is a diagonal matrix and Pᵗ $P \neq = I_{3}$ .✓
C.One of the eigenvalues of A is not real.
D.A has only real eigenvalues and it is not diagonalizable over $R$ .

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.

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50 are analysed free — try those first.

Related counterexample: Diagonalisable ⇒ invertible

Orthogonally diagonalizable means symmetric

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