Which of the following values of a satisfy the equation \left|\right. \left(\left(\right. 1 + \alpha \left.\right)\right)^{2} & \left(\left(\right. 1 + 2 \alpha \left.\right)\right)^{2} & \left(\left(\right. 1 + 3 \alpha \left.\right)\right)^{2} \\ \left(\left(\right. 2 + \alpha \left.\right)\right)^{2} & \left(\left(\right. 2 + 2 \alpha \left.\right)\right)^{2} & \left(\left(\right. 2 + 3 \alpha \left.\right)\right)^{2} \\ \left(\left(\right. 3 + \alpha \left.\right)\right)^{2} & \left(\left(\right. 3 + 2 \alpha \left.\right)\right)^{2} & \left(\left(\right. 3 + 3 \alpha \left.\right)\right)^{2} \left|\right. = \textrm{ }

Engineering

Mathematics

Multiplication of Determinants

Question

Which of the following values of a satisfy the equation $| \begin{matrix} {(1 + α)}^{2} & {(1 + 2 α)}^{2} & {(1 + 3 α)}^{2} \\ {(2 + α)}^{2} & {(2 + 2 α)}^{2} & {(2 + 3 α)}^{2} \\ {(3 + α)}^{2} & {(3 + 2 α)}^{2} & {(3 + 3 α)}^{2} \end{matrix} | = - 648 α$ ?

–4

–9

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$| \begin{matrix} 1 & 1 & 1 \\ 4 & 2 & 1 \\ 9 & 3 & 1 \end{matrix} | | \begin{matrix} 1 & 2 α & α^{2} \\ 1 & 4 α & 4 α^{2} \\ 1 & 6 α & 9 α^{2} \end{matrix} | = - α 3 {| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \end{matrix} |}^{2} = - α 3 {| \begin{matrix} 1 & 1 & 1 \\ 0 & 1 & 3 \\ 0 & 1 & 5 \end{matrix} |}^{2}$

⇒ 4a³ = – 648α ⇒ α = 0, α² = 81 ⇒ α = ± 9.