If a , b , c , λ ∈ N , then the least possible value of a 2 + λ a b a c b a b 2 + λ b c c a c b c 2 + λ is

Step-by-step solution

Multiplying first, second, third rows by a , b , c and then taking a , b , c common from first, second, third columns, we get, D = a 2 + λ a 2 a 2 b 2 b 2 + λ b 2 c 2 c 2 c 2 + λ Now, R 1 ↔ R 1 + R 2 + R 3 = a 2 + b 2 + c 2 + λ 1 1 1 b 2 b 2 + λ b 2 c 2 c 2 c 2 + λ Now, C 2 ↔ C 2 - C 1   ,   C 3 → C 3 - C 1 = a 2 + b 2 + c 2 + λ 1 0 0 b 2 λ 0 c 2 0 λ = λ 2 a 2 + b 2 + c 2 + λ Hence, D m i n = 4 when a = b = c = λ = 1