Consider a matrix A = α β γ α 2 β 2 γ 2 β + γ γ + α α + β , where α , β , γ are three distinct natural numbers. If det adj ( adj adj adj A α - β 16 β - γ 16 γ - α 16 = 2 32 × 3 16 , then the number of such 3 - tuples α , β , γ is _______.

Step-by-step solution

Given A = α β γ α 2 β 2 γ 2 β + γ γ + α α + β R 3 → R 3 + R 1 A = α β γ α 2 β 2 γ 2 α + β + γ α + β + γ α + β + γ ⇒ A = α + β + γ α β γ α 2 β 2 γ 2 1 1 1 ⇒ A = α + β + γ α - β β - γ γ - α We know adj A = A n - 1 ⇒ adj adj A = A n - 1 2 ⇒ adj adj adj adj A = A n - 1