If $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&1&{ - 2}\\2&1&{ - 3}\\5&4&{ - 9}\end{array}} \right]$, find $|A|.$
If $A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&1&{ - 2}\\2&1&{ - 3}\\5&4&{ - 9}\end{array}} \right]$, find $|A|.$
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$A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&1&{ - 2}\\2&1&{ - 3}\\5&4&{ - 9}\end{array}} \right]$
$\Rightarrow$ $|A| = \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&1&{ - 2}\\2&1&{ - 3}\\5&4&{ - 9}\end{array}} \right| = 1\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}1&{ - 3}\\4&{ - 9}\end{array}} \right| - 1\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&{ - 3}\\5&{ - 9}\end{array}} \right| - 2\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}2&1\\5&4\end{array}} \right|$
$= 1\left( { - 9 + 12} \right) - 1\left( {18 + 15} \right) - 2\left( {8 - 5} \right) = 3 + 3 - 6 = 0.$
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