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Evaluate \(\left| {\begin{array}{*{20}{c}} 3&{ - 4}&5 \\ 1&1&{ - 2} \\ 2&3&1 \end{array}} \right|\)
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Evaluate \(\left| {\begin{array}{*{20}{c}} 3&{ - 4}&5 \\ 1&1&{ - 2} \\ 2&3&1 \end{array}} \right|\)

Answer

Given: \(\left| {\begin{array}{*{20}{c}} 3&{ - 4}&5 \\ 1&1&{ - 2} \\ 2&3&1 \end{array}} \right|\)
Expanding along first row, \(3\left| {\begin{array}{*{20}{c}} 1&{ - 2} \\ 3&1 \end{array}} \right| - \left( { - 4} \right)\left| {\begin{array}{*{20}{c}} 1&{ - 2} \\ 2&1 \end{array}} \right| + 5\left| {\begin{array}{*{20}{c}} 1&1 \\ 2&3 \end{array}} \right|\)
= 3(1 + 6) + 4{1 - ( - 4)} + 5(3 - 2)
\( = 3 \times 7 + 4 \times 5 + 5 \times 1\)
= 21 + 20 + 5 = 46
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