Consider a matrix A = [ 2 2 2 ] where are three distinct natural numbers. If det ( adj ( adj ( adj ( adj A ) ) ) ) 16 16 16 = 2 32 × 3 16 , then the number of such 3-triples is

Question

Consider a matrix $A =$ $[\begin{matrix} α & β & λ \\ α^{2} & β^{2} & λ^{2} \\ β + λ & λ + α & α + β \end{matrix}]$ where $α, β, λ$ 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-triples $(α, β, γ)$ is _______ .

Smart Achievers · Accepted Answer

(42)

$Given matrix is A =$ $[\begin{matrix} α & β & γ \\ α^{2} & β^{2} & γ^{2} \\ β + γ & γ + α & α + β \end{matrix}]$

$R_{3} \to R_{3} + R_{1}$

$\Rightarrow$ $| A |$ $=$ $| α + β + γ |$ $| \begin{matrix} α & β & γ \\ α^{2} & β^{2} & γ^{2} \\ 1 & 1 & 1 \end{matrix} |$

$\Rightarrow$ $| A |$ $= (α + β + γ) (α - β) (β - γ) (γ - α)$

$∵$ $| adj A |$ $=$ $| A |^{n - 1}$

$| adj (adj A) |$ $=$ $| A |^{{(n - 1)}^{2}}$

$| adj (adj (adj A)) |$ $=$ $| A |^{{(n - 1)}^{3}}$ $=$ $| A |^{8}$

$| adj (adj (adj (adj A))) |$ $=$ $| A |^{{(n - 1)}^{4}}$ $=$ $| A |^{16}$

$∴$ $(α + β + γ) = 2^{32} \cdot 3^{16}$

Solution

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Solution

Q. Consider a matrix A=[αβλα2β2λ2β+λλ+αα+β] where α,β,λ are three distinct natural numbers. If det(adj(adj(adj(adjA))))(α-β)16(β-γ)16(γ-α)16=232×316, then the number of such 3-triples (α,β,γ) is _______ .

Ans. (42) Given matrix is A=[αβγα2β2γ2β+γγ+αα+β] R3→R3+R1 ⇒|A|=|α+β+γ||αβγα2β2γ2111| ⇒|A|=(α+β+γ)(α-β)(β-γ)(γ-α) ∵ |adjA|=|A|n-1 |adj(adjA)|=|A|(n-1)2 |adj(adj(adjA))|=|A|(n-1)3=|A|8 |adj(adj(adj(adjA)))|=|A|(n-1)4=|A|16 ∴ (α+β+γ)=232·316

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