If = | array lll 1 & a & a^2 1 & b & b^2 1 & c & c^2 array | and ₁= | array ccc 1 & 1 & 1 b c & c a & a b a & b & c array | , then

Step-by-step solution

Given, = | array lll 1 & a & a^2 1 & b & b^2 1 & c & c^2 array | and ₁= | array ccc 1 & 1 & 1 b c & c a & a b a & b & c array | = | array lll 1 & a & a^2 1 & b & b^2 1 & c & c^2 array | aligned R₁ R₁-R₂ and R₂ R₂-R₃ = | array ccc 0 & a-b & a^2-b^2 0 & b-c & b^2-c^2 1 & c & c^2 array | =(a-b)(b-c) | array ccc 0 & 1 & a+b 0 & 1 & b+c 1 & c & c^2 array | =(a-b)(b-c)(c-a) aligned aligned & Now, ₁= | array ccc 1 & 1 & 1 b c & c a & a b a & b & c array | & C₁ C₁-C₂, C₂ C₂-C₃ aligned ₁= | array ccc 0 & 0 & 1 c(b-a) & a(c-