Evaluate the following determinants without expansion as far as possible.
$∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 3 5 & 5 & 7 8 & 14 & 20 \end{matrix} ∣ ∣ ∣ ∣$

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Solution

$∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 3 5 & 5 & 7 8 & 14 & 20 \end{matrix} ∣ ∣ ∣ ∣$

Applying $R_{3} \to R_{3} - R_{2} - 3 R_{1}$
we get
$∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 3 5 & 5 & 7 0 & 3 & 4 \end{matrix} ∣ ∣ ∣ ∣$

Applying $R_{2} \to R_{2} - R_{1} - R_{3}$
we get
$∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 3 4 & 0 & 0 0 & 3 & 4 \end{matrix} ∣ ∣ ∣ ∣$

Expanding second row,
$- 4 * (2 * 4 - 3 * 3) + 0 - 0 = - 4 (8 - 9) = 4$

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