If a, b, c are distinct positive real numbers, then the value of the determinant | array lll a & b & c b & c & a c & a & b array | is

Step-by-step solution

aligned & Let A = | array lll a & b & c b & c & a c & a & b array | [ C ₁ C ₁ C ₂ C ₃ ] & = | array lll a+b+c & b & c a+b+c & c & a a+b+c & a & b array | & =(a+b+c) | array lll 1 & b & c 1 & c & a 1 & a & b array | & [ R ₂ R ₂- R ₁ and R ₃ R ₃- R ₁ ] & =(a+b+c) | array ccc 1 & b & c 0 & c-b & a-c 0 & a-b & b-c array | & =(a+b+c)[(c-b)(b-c)-(a-c)(a-b)] & =(a+b+c) [b c-b^2-c^2+b c-a^2+a b+a c-a b c ] & =-(a+b+c) (a^2+b^2+c^2-a b-b c-c a ) & =- 1 2 (a+b+c) [(a-b)^2+(b-c)^2+(c-a)^2 ] & aligned Since it is given that a,