QUESTION

# Expand the determinant: $\left| {\begin{array}{*{20}{c}} 2&{ - 2}&5 \\ 4&6&{ - 2} \\ 3&{ - 4}&1 \end{array}} \right|$ .

Hint-In this question, we use the concept of property of determinants. We apply some row and column operation to simplify determinants in easy form then after simplify we can easily expand the determinants.
We have a determinant $\left| {\begin{array}{*{20}{c}} 2&{ - 2}&5 \\ 4&6&{ - 2} \\ 3&{ - 4}&1 \end{array}} \right|$ and we have to expand it. So, before expanding we can simplify determinant by applying some row and column operation then we easily expand the determinant.
We apply, ${C_2} \to {C_2} + {C_1}$
$\Rightarrow \left| {\begin{array}{*{20}{c}} 2&0&5 \\ 4&{10}&{ - 2} \\ 3&{ - 1}&1 \end{array}} \right|$
Apply, ${C_3} \to {C_3} + {C_2}$
$\Rightarrow \left| {\begin{array}{*{20}{c}} 2&0&5 \\ 4&{10}&8 \\ 3&{ - 1}&0 \end{array}} \right|$
Apply, ${R_2} \to {R_2} + \left( { - 2} \right){R_1}$
$\Rightarrow \left| {\begin{array}{*{20}{c}} 2&0&5 \\ 0&{10}&{ - 2} \\ 3&{ - 1}&0 \end{array}} \right|$
$\Rightarrow 2\left[ {10 \times 0 - \left( { - 2} \right) \times \left( { - 1} \right)} \right] - 0\left[ {0 - \left( { - 2} \right) \times 3} \right] + 5\left[ {0 \times \left( { - 1} \right) - 10 \times 3} \right] \\ \Rightarrow 2\left( { - 2} \right) - 0 + 5\left( { - 30} \right) \\ \Rightarrow - 4 - 150 \\ \Rightarrow - 154 \\$
So, the answer of determinant $\left| {\begin{array}{*{20}{c}} 2&{ - 2}&5 \\ 4&6&{ - 2} \\ 3&{ - 4}&1 \end{array}} \right|$ is -154.