If | array ccc -2 a & a+b & a+c b+a & -2 b & b+c c+a & b+c & -2 c array | = (a+b() b+c() c+a) 0 then is equal to

Step-by-step solution

Let = | array ccc -2 a & a+b & a+c b+a & -2 b & b+c c+a & b+c & -2 c array | Applying C₁+C₃ and C₂+C₃ = | array ccc -a+c & 2 a+b+c & a+c 2 b+a+c & -b+c & b+c a-c & b-c & -2 c array | Now, applying R₁+R₃ and R₂+R₃ = | array ccc 0 & 2(a+b & ) a-c 2(a+b & 0 & b-c a-c & b-c & -2 c array | On expanding, we get aligned = & -2(a+b) -2 c[2(a+b)] & -(a-c)(b-c) & +(a-c)[2(a+b)(b-c)] = & 8 c (a+b)(a+b) & +4(a+b)(a-c)(b-c) aligned aligned & =4(a+b) [2 a c+2 b c+a b-b c-a c+c^2 ] & =4(a+b) [a c+b c+a b+c^2 ] & =4(a+b)[c(a+c)+b(