Evaluate: | * 20 c x&y& x + y y& x + y &x x+ y &x&y | - Vidyadip

Question

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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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
$= 2\left( {x + y} \right).1\left| {\begin{array}{*{20}{c}} x&{x - y} \\ { - y}&{ - x} \end{array}} \right|$
$= 2(x + y){ -x^2 + y(x - y)}$
$= 2(x + y)(-x^2 + xy - y^2)$
$= -2(x + y)(x^2 - xy + y^2)$
$= -2(x^3 + y^3)$

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