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

Answer

\(\left| {\begin{array}{*{20}{c}} 2&7&{65} \\ 3&8&{75} \\ 5&9&{86} \end{array}} \right|\)
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]
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