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Properties of Determinants

Find the solu...

Question

Find the solution set of $∣ ∣ ∣ ∣ \begin{matrix} 2 + x & 2 - x & 2 - x 2 - x & 2 + x & 2 - x 2 - x & 2 - x & 2 + x \end{matrix} ∣ ∣ ∣ ∣ = 0$

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Solution

The value of the determinant $A = ⎛ ⎜ ⎝ \begin{matrix} 2 + x & 2 - x & 2 - x 2 - x & 2 + x & 2 - x 2 - x & 2 - x & 2 + x \end{matrix} ⎞ ⎟ ⎠$ is $(2 + x) [(2 + x)^{2} - (2 - x)^{2}] - (2 - x) [4 - x^{2} - 4 - x^{2} + 4 x] + (2 - x) [4 + x^{2} - 4 x - 4 + x^{2}] = 0$
$16 x + 8 x^{2} + 16 x^{2} - 16 x - 4 x^{3} = 0$
$- 4 x^{3} + 24 x^{2} = 0$
$x^{2} (- 4 + 24 x) = 0$
$\Rightarrow x = 0, 6$
Therefore the solution set is ${0, 6}$

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