If a, b, c , are non zero complex numbers satisfying a^2+b^2+c^2=0 and | array ccc b^2+c^2 & a b & a c a b & c^2+a^2 & b c a c & b c & a^2+b^2 array |=k a^2 b^2 c^2 , then k is equal to

Step-by-step solution

Let = | array ccc b^2+c^2 & a b & a c a b & c^2+a^2 & b c a c & b c & a^2+b^2 array | Multiply C₁ by a, C₂ by b and C₃ by c and hence divide by a b c . = 1 a b c | array ccc a (b^2+c^2 . & .a b^2 ) & a c^2 a^2 b & b (c^2+a^2 . & .b c^2 ) a^2 c & b^2 c & c (a^2+b^2 . array | Take out a, b, c common from R₁, R₂ and R₃ respectively. aligned & = a b c a b c | array ccc b^2+c^2 & b^2 & c^2 a^2 & c^2+a^2 & c^2 a^2 & b^2 & a^2+b^2 array | & Apply C₁ C₁-C₂-C₃ & = | array ccc 0 & b^2 & c^2 -2 c^2 & c^2+a^2 & c^2 -2 b^2 & b^