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

Question

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 _______.

TestHub · Accepted Answer

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 4 Here adj adj adj adj A = A 2 4 = A 16 Given adj ( adj adj adj A α - β 16 β - γ 16 γ - α 16 = 2 32 × 3 16 ⇒ α + β + γ 16 α - β 16 β - γ 16 γ - α 16 α - β 16 β - γ 16 γ - α 16 = 2 32 × 3 16 ∴ α + β + γ 16 = 2 32 · 3 16 ⇒ α + β + γ 16 = 12 16 ⇒ α + β + γ = 12 ∵ α , β , γ ∈ N α - 1 + β - 1 + γ - 1 = 9 Possible number all tuples α , β , γ will be C 2 11 = 55 1 case for α = β = γ and 12 case when any two of these are equal are also included here but α ≠ β ≠ γ Hence, number of distinct tuples α , β , γ = 55 - 13 = 42

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