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Evaluate the determinant \(\left| {\begin{array}{*{20}{c}} {{x^2} - x + 1}&{x - 1} \\ {x + 1}&{x + 1} \end{array}} \right|\)
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Question
MathsClass 12

Evaluate the determinant \(\left| {\begin{array}{*{20}{c}} {{x^2} - x + 1}&{x - 1} \\ {x + 1}&{x + 1} \end{array}} \right|\)

Answer

\( x ^{3} - x ^{2} + 2\)
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Solution

Applying \(C_1\rightarrow C_1-C_2,we\ get,\)
.\(\left| {\begin{array}{*{20}{c}} {{x^2} - x + 1}&{x - 1} \\ {x + 1}&{x + 1} \end{array}} \right| = \left| {\begin{array}{*{20}{c}} {{x^2} - 2x + 2}&{x - 1} \\ 0&{x + 1} \end{array}} \right|\)
\(= \left( {{x^2} - 2x + 2} \right).\left( {x + 1} \right) - \left( {x - 1} \right).0\)
\(= x ^{3} - 2x ^{2} + 2x + x ^{2} - 2x + 2\)
\(= x ^{3} - x ^{2} + 2\)
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