Let for i =1,2,3, p _ i ( x ) be a polynomial of degree 2 in x, p^ _i(x) and p^ _i(x) be the first and second order derivatives of p_i(x) respectively. Let, . A ( x )= [ array lll p ₁( x ) & p ₁^ ( x ) & p ₁^ x ( p ₂( x ) & p ₂^ ( x ) & p ₂^ ( p ₃( x ) & p ₃^ ( x ) & p ₃^ ( x array ] ) and B ( x )=[ A ( x )]^ T A ( x ) . Then determinant of B ( x ) :

Step-by-step solution

Let p₁ x=a₁ x^2+b₁ x+c₁ p₂ x=a₂ x^2+b₂ x+c₂ and p₃ x=a₃ x^2+b₃ x+c₃ where a₁, a₂, a₃, b₁, b₂, b₃, c₁, c₂, c₃ are real numbers. gathered A (x)= [ array ccc a₁ x^2+b₁ x+c₁ & 2 a₁ x+b₁ & 2 a₁ a₂ x^2+b₂ x+c₂ & 2 a₂ x+b₂ & 2 a₂ a₃ x^2+b₃ x+c₃ & 2 a₃ x+b₃ & 2 a₃ array ] = [ array llll a₁ x^2+b₁ x+c₁ & a₂ x^2+b₂ x+c₂ & a₃ x^2+b₃ x+c₃ 2 a₁ x+b₁ & 2 a₂ x+b₂ & 2 a₃ x+b₂ 2 a₁ & 2 a₂ & 2 a₃ array ] [ array ccc a₁ x^2+b₁ x+c₁ & 2 a₁ x+b₁ & 2 a₁ a₂ x^2+b₂ x+c₂ & 2 a₂ x+b₂ & 2 a₂ a₃ x^2+b₃ x+c₃ & 2 a₃ x+b₃ & 2 a₃ array ] gathered