First slide

Introduction to Determinants

Question

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

Moderate

A

9
B

-4
C

-9
D

9

Solution

$\begin{array}{l} |\begin{matrix} {(1 + α)}^{2} & {(1 + 2 α)}^{2} & {(1 + 3 α)}^{2} \\ {(2 + α)}^{2} & {(2 + 2 α)}^{2} & {(2 + 3 α)}^{2} \\ {(3 + α)}^{2} & {(3 + 2 α)}^{2} & {(3 + 3 α)}^{2} \end{matrix}| = - 648 α • R_{2} \to R_{2} - R_{1}, . R_{3} \to R_{3} - R_{2} \\ |\begin{matrix} {(1 + α)}^{2} & {(1 + 2 α)}^{2} & {(1 + 3)}^{2} \\ 3 + 2 α & 3 + 4 α & 3 + 6 α \\ 5 + 2 α & 5 + 4 α & 5 + 6 α \end{matrix}| = - 648 α • R_{3} \to R_{3} - R_{2} \\ |\begin{matrix} {(1 + α)}^{2} & {(1 + 2 α)}^{2} & {(1 + 3 α)}^{2} \\ 3 + 2 α & 3 + 4 α & 3 + 6 α \\ 2 & 2 & 2 \end{matrix}| = - 648 α \\ |\begin{matrix} {(1 + α)}^{2} & {(1 + 2 α)}^{2} & {(1 + 3 α)}^{2} \\ 3 + 2 α & 3 + 4 α & 3 + 6 α \\ 1 & 1 & 1 \end{matrix}| = - 324 α • C_{2} \to C_{2} - C_{1}; C_{3} \to C_{3} - C_{2} \\ |\begin{matrix} {(1 + α)}^{2} & 2 α + 3 α^{2} & 2 α + 5 α^{2} \\ 3 + 2 α & 2 α & 2 α \\ 1 & 0 & 0 \end{matrix}| = - 324 α • C_{3} \to C_{3} - C_{2} \\ |\begin{matrix} {(1 + α)}^{2} & (2 + 3 α) α & 2 α^{2} \\ 3 + 2 α & 2 α & 0 \\ 1 & 0 & 0 \end{matrix}| = - 324 α \\ 2 α (- 2 α^{2}) = - 324 α \\ \Rightarrow α^{2} = \frac{- 324}{- 4} = 81 \Rightarrow α = \pm 9 \end{array}$

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