Using the property of determinants and without expanding, prove that

$∣ ∣ ∣ ∣ \begin{matrix} 2 & 7 & 65 3 & 8 & 75 5 & 9 & 86 \end{matrix} ∣ ∣ ∣ ∣ = 0$

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Solution

Let $Δ = ∣ ∣ ∣ ∣ \begin{matrix} 2 & 7 & 65 3 & 8 & 75 5 & 9 & 86 \end{matrix} ∣ ∣ ∣ ∣$
Applying $C_{1} = C_{1} + 9 C_{2}$ , we get,
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} 65 & 7 & 65 75 & 8 & 75 86 & 9 & 86 \end{matrix} ∣ ∣ ∣ ∣ = 0$
Since first and third column are same.

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