Question
Without expanding, evaluate the following determinants :

$\left|\begin{array}{lll}2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86\end{array}\right|$

Answer

Let $D=\left|\begin{array}{lll}2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86\end{array}\right|$

Applying $\mathrm{Cx}_3-\mathrm{C}_3-9 \mathrm{C}_2$, we get

$D=\left|\begin{array}{lll}2 & 7 & 2 \\ 3 & 8 & 3 \\ 5 & 9 & 5\end{array}\right|$

$=0 \ldots\left[\because C_1\right.$ and $C_3$ are identical $]$

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