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# Question and Answer

Using the property of determinant and without expanding prove that $$\left| {\begin{array}{*{20}{c}} 2&7&{65} \\ 3&8&{75} \\ 5&9&{86} \end{array}} \right| = 0$$
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## QuestionMathsClass 12

Using the property of determinant and without expanding prove that $$\left| {\begin{array}{*{20}{c}} 2&7&{65} \\ 3&8&{75} \\ 5&9&{86} \end{array}} \right| = 0$$

$$\left| {\begin{array}{*{20}{c}} 2&7&{65} \\ 3&8&{75} \\ 5&9&{86} \end{array}} \right|$$
Operating $${C_3} \to {C_3} - {C_1}$$
$$= \left| {\begin{array}{*{20}{c}} 2&7&{63} \\ 3&8&{72} \\ 5&9&{81} \end{array}} \right|$$
Taking 9 common from third column,
$$= 9\left| {\begin{array}{*{20}{c}} 2&7&7 \\ 3&8&8 \\ 5&9&9 \end{array}} \right|$$
$$= 9 \times 0 = 0$$ [ two columns are identical]