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 (R^3 . means all the three components of the vector are real) Question: The matrix A is

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 c a₁₁ x₁+a₁₂ x₂ a₂₁ x₁+a₂₂ x₂ x₃+a₂₃ x₃ a₃₁ x₁+a₃₂ x₂+a₃₃ x₃ array ] Since A x is orthogonal to x for every x in R^3 , so aligned & 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