Using the property of determinants and without expanding, prove t… - Vidyadip

Question

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

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Answer

$\text{On}\begin{vmatrix}2&7&65\\3&8&75\\5&9&86\end{vmatrix},\ \text{operating}\ \text{C}_3\rightarrow\text{C}_3-\text{C}_1$
$=\begin{vmatrix}2&7&63\\3&8&72\\5&9&81\end{vmatrix}=9\begin{vmatrix}2&7&7\\3&8&8\\5&9&9\end{vmatrix}=9\times0=0$ $\left[\because\text{two columns are identical}\right]$ Proved.

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