If A = [ * 20 c 1&1& - 2 2&1& - 3 5&4&-9 ] find |A|. - Vidyadip

Question

If $A = \left[ {\begin{array}{*{20}{c}} 1&1&{ - 2} \\ 2&1&{ - 3} \\ 5&4&-9 \end{array}} \right]$ find |A|.

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Answer

Given: $A = \left[ {\begin{array}{*{20}{c}} 1&1&{ - 2} \\ 2&1&{ - 3} \\ 5&4&9 \end{array}} \right]$
$ \Rightarrow \left| A \right| = \left| {\begin{array}{*{20}{c}} 1&1&{ - 2} \\ 2&1&{ - 3} \\ 5&4&9 \end{array}} \right|$
Expanding along first row, $1\left| {\begin{array}{*{20}{c}} 1&{ - 3} \\ 4&{ - 9} \end{array}} \right| - 1\left| {\begin{array}{*{20}{c}} 2&{ - 3} \\ 5&{ - 9} \end{array}} \right| + \left( { - 2} \right)\left| {\begin{array}{*{20}{c}} 2&1 \\ 5&4 \end{array}} \right|$
= {-9(-12)} - {-18 - (-15)} - 2(8 - 5)
= -9 + 12 - (-18 + 15) -2(3)
= 3(-3) - 6
= 3 + 3 - 6 = 0

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