If α, β ≠ 0, and f (n) = αn + βn and = K (1 – α)2 (1 – β)2 (α

Engineering

Mathematics

Multiplication of Determinants

Question

If α, β ≠ 0, and f (n) = αⁿ + βⁿ and $| \begin{matrix} 3 & 1 + f (1) & 1 + f (2) \\ 1 + f (1) & 1 + f (2) & 1 + f (3) \\ 1 + f (2) & 1 + f (3) & 1 + f (4) \end{matrix} |$ = K (1 – α)² (1 – β)² (α – β)², then K is equal to

αβ

$\frac{1}{αβ}$

– 1

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$| \begin{matrix} 1 & 1 & 1 \\ 1 & α & β \\ 1 & α^{2} & β^{2} \end{matrix} | | \begin{matrix} 1 & 1 & 1 \\ 1 & α & β \\ 1 & α^{2} & β^{2} \end{matrix} |$

(Using product of two determinants)

= (1 – α)² (α – β)² (β – 1)²

k = 1