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Evaluate: \(\left| {\begin{array}{*{20}{c}} x&y&{x + y} \\ y&{x + y}&x \\ {x+ y}&x&y \end{array}} \right|\)
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Evaluate: \(\left| {\begin{array}{*{20}{c}} x&y&{x + y} \\ y&{x + y}&x \\ {x+ y}&x&y \end{array}} \right|\)

Answer

Let \(\Delta = \left| {\begin{array}{*{20}{c}} x&y&{x + y} \\ y&{x + y}&x \\ {x + y}&x&y \end{array}} \right|\)
\(\left[ {{R_1} \to {R_1} + {R_2} + {R_3}} \right]\)
\(= \left| {\begin{array}{*{20}{c}} {2\left( {x + y} \right)}&{2\left( {x + y} \right)}&{2\left( {x + y} \right)} \\ y&{x + y}&x \\ {x + y}&x&y \end{array}} \right|\)
Taking 2(x+y) common from first row
\(= 2\left( {x + y} \right)\left| {\begin{array}{*{20}{c}} 1&1&1 \\ y&{x + y}&x \\ {x + y}&x&y \end{array}} \right|\)
\(\left[ {{C_2} \to {C_2} - {C_1}and\,\,{C_3} \to {C_3} - {C_1}} \right]\)
\( = 2\left( {x + y} \right)\left| {\begin{array}{*{20}{c}} 1&0&0 \\ y&{x + y - y}&{x - y} \\ {x + y}&{x - x - y}&{y - x - y} \end{array}} \right|\)
\(= 2\left( {x + y} \right)\left| {\begin{array}{*{20}{c}} 1&0&0 \\ y&x&{x - y} \\ {x + y}&{ - y}&{ - x} \end{array}} \right|\)
Expanding along Ist row
\(= 2\left( {x + y} \right).1\left| {\begin{array}{*{20}{c}} x&{x - y} \\ { - y}&{ - x} \end{array}} \right|\)
= 2(x + y){ -x2 + y(x - y)}
= 2(x + y)(-x2 + xy - y2)
= -2(x + y)(x2 - xy + y2)
= -2(x3 + y3)
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