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Let \(\rm A = \begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&1&1\\\ 0&0&1&1\end{pmatrix}\) and consider the symmetric bilinear form on R4 given by (v, w) = vt Aw, for v, w ∈ ℝ4. Which of the following statements is true? 

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  1. A is invertible
  2. There exist non-zero vectors v, w such that 〈v, w〉 = 0
  3. 〈u, v〉 ≠ 〈u, w〉 for all non-zero vectors u, v, w with v ≠ w
  4. Every eigenvalue of A2 is positive

Answer (Detailed Solution Below)

Option 2 : There exist non-zero vectors v, w such that 〈v, w〉 = 0
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The Correct answer is (2).

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Last updated on Jun 5, 2025

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