Consider a matrix A=αβγα2β2γ2β+γγ+αα+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=232×316, then the number of such 3

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Consider a matrix $A = [\begin{array}{ccc} α & β & γ \\ α^{2} & β^{2} & γ^{2} \\ β + γ & γ + α & α + β \end{array}],$ where $α, β, γ$ are three distinct natural numbers.

If $\frac{\det (adj (adj (adj (adj A))))}{(α - β)^{16} (β - γ)^{16} (γ - α)^{16}} = 2^{32} \times 3^{16},$ then the number of such 3 – tuples ( $α, β, γ$ ) is _______ .

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answer is 42.

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Detailed Solution

$\begin{array}{l} A = [\begin{array}{ccc} α & β & γ \\ α^{2} & β^{2} & γ^{2} \\ β + γ & γ + α & α + β \end{array}] \\ R_{3} \to R_{3} + R_{1} \\ \Rightarrow | A | = | α + β + γ | |\begin{array}{lcc} α & β & γ \\ α^{2} & β^{2} & γ^{2} \\ 1 & 1 & 1 \end{array}| \\ \Rightarrow | A | = (α + β + γ) (α - β) (β - γ) (γ - α) \\ ∵ | adj A | = | A |^{n - 1} \\ | adj (adj A) | = | A |^{(n - 1)^{2}} \\ | adj (adj (adj (adj A))) | = | A |^{(n - 1)^{4}} = | A |^{2^{4}} = | A |^{16} \\ ∴ (α + β + γ)^{16} = 2^{32} \cdot 3^{16} \\ \Rightarrow (α + β + γ)^{16} = {(2^{2} \cdot 3)}^{16} = (12)^{16} \\ \Rightarrow α + β + γ = 12 \\ ∵ α, β, γ \in N \\ (α - 1) + (β - 1) + (γ - 1) = 9 \end{array}$