If = | array lll 1 & a & a^2 1 & b & b^2 1 & c & c^2 array |=k(a-b)(b-c)(c-a) , then k is equal to

Step-by-step solution

aligned & = | array ccc 1 & a & a^2 1 & b & b^2 1 & c & c^2 array |= | array ccc 1 & a & a^2 0 & b-a & b^2-a^2 0 & c-a & c^2-a^2 array | & ( using R₂ R₂-R₁ and R₃ R₃-R₁ ) & =(b-a)(c-a) | array ccc 1 & a & a^2 0 & 1 & b+a 0 & 1 & c+a array | & =(b-a)(c-a)(c+a-b-a) & (b-a)(c-a)(c-b) & =k(a-b)(b-c)(c-a)(given) & k=1 aligned