Using the properties of determinant and without expanding , prove that:
$∣ ∣ ∣ ∣ \begin{matrix} 2 & 7 & 65 3 & 8 & 75 5 & 9 & 86 \end{matrix} ∣ ∣ ∣ ∣ = 0$

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Solution

$∣ ∣ ∣ ∣ \begin{matrix} 2 & 7 & 65 3 & 8 & 75 5 & 9 & 86 \end{matrix} ∣ ∣ ∣ ∣ R_{1} \to R_{1} - R_{2} R_{2} \to R_{2} - R_{3} R_{3} \to R_{3} - R_{2} ∣ ∣ ∣ ∣ \begin{matrix} - 1 & - 1 & - 10 - 2 & - 1 & - 11 2 & 1 & 11 \end{matrix} ∣ ∣ ∣ ∣ R_{1} \to R_{1} - R_{2} R_{2} \to R_{2} + R_{3} ∣ ∣ ∣ ∣ \begin{matrix} 1 & 0 & 1 0 & 0 & 0 - 2 & 1 & 11 \end{matrix} ∣ ∣ ∣ ∣ ∴ 1 (0 - 0) - 0 + 1 (0 - 0) = 0$

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