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Paragraph: Let A be a 3 3 matrix given by A= [a_ i j ] such that the vector A x is orthogonal to x for every non zero x in R^3 . Question: The matrix A is

Options

  1. ASingular
  2. BNon singular
  3. CSymmetric
  4. DSkew symmetric

Correct answer

D. Skew symmetric

Step-by-step solution

Let A= [ array lll a₁₁ & a₁₂ & a₁₃ a₂₁ & a₂₂ & a₂₃ a₃₁ & a₃₂ & a₃₃ array ] and x =x₁ i +x₂ j +x₃ k A x = [ array lll a₁₁ & a₁₂ & a₁₃ a₂₁ & a₂₂ & a₂₃ a₃₁ & a₃₂ & a₃₃ array ] [ array l x₁ x₂ x₃ array ]= [ array l a₁₁ x₁+a₁₂ x₂+a₁₃ x₃ a₂₁ x₁+a₂₂ x₂+a₂₃ x₃ a₃₁ x₁+a₃₂ x₂+a₃₃ x₃ array ] Since A x is orthogonal to x for every x in R^3 , so A x x =0 (a₁₁ x₁+a₁₂ x₂+a₁₃ x₃ ) x₁+ (a₂₁ x₁+a₂₂ x₂+a₂₃ x₃ ) x₂+ (a₃₁ x₁+a₃₂ x₂+a₃₃ x₃ ) x₃=0 (a₁₁ x₁^2+a₂₂ x₂^2+a₃₃ x₃^2 )+ (a₁₂+a₂₁ ) x₁ x₂+ (a₁₃+a₃₁ ) x₁ x₃+ (a₂₃+a₃₂ ) x₂ x₃=0 ...(1

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