If | array ccc a+b+2 c & a & b c & 2 a+b+c & b c & a & a+2 b+c array |=2 , then a^3+b^3+c^3-3 a b c=

Step-by-step solution

aligned & | array ccc a+b+2 c & a & b c & 2 a+b+c & b c & a & a+2 b+c array |=2 & 2(a+b+c) | array ccc 1 & a & b 1 & b+c+2 a & b 1 & a & c+a+2 b array |=2 aligned [Applying C₁ C₁+C₂+C₃ and taking 2(a+b+c) common from .C₁ ] 2(a+b+c) | array ccc 1 & a & b 0 & b+c+a & 0 0 & 0 & c+a+b array |=2 [Applying R₂ R₂-R₁ and R₃ R₃-R₁ ] aligned & 2(a+b+c)^3=2 [expanding along C₁ ] & (a+b+c)^3=1 a+b+c=1 & aligned Now, a^3+b^3+c^3-3 a b c aligned & =(a+b+c) (a^2+b^2+c^2-a b-b c-c a ) & =1 [a^2+b^2+c^2-a b-b c-c a ] & =(a+b+c)^2-2