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Part CCSIR NET December 2023

Quadratic forms, positive definiteness, Sylvester's law

Let A be a real diagonal matrix with characteristic polynomial λ3−2λ2−λ+2\lambda^{3} - 2\lambda^{2} - \lambda + 2. Define a bilinear form ⟨v, w⟩ = vᵗAw on R3\mathbb{R}^{3}. Which of the following statements are true?

  1. A.A is positive definite.
  2. B.A2A^{2} is positive definite.✓ Correct
  3. C.There exists a nonzero v∈R3v \in \mathbb{R}^{3} such that ⟨v, v⟩ = 0.✓ Correct
  4. D.rank A = 2.

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: Every real symmetric matrix is positive definite if det > 0

More on this topic

The chapter behind this: Quadratic forms, signature and definiteness — free to read

From Inner Product Spaces and Forms › Quadratic forms, positive definiteness, Sylvester's law

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