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$\left|\begin{array}{ccc}x & y & x+y \\ y & x+y & x \\ x+y & x & y\end{array}\right|$ નું મૂલ્ય શોધો.
$2\left(x^{3}+y^{3}\right)$
$2\left(x^{3}-y^{3}\right)$
$-2\left(x^{3}-y^{3}\right)$
$-2\left(x^{3}+y^{3}\right)$
Solution
$\Delta=\left|\begin{array}{ccc}x & y & x+y \\ y & x+y & x \\ x+y & x & y\end{array}\right|$
Applying $R_{1} \rightarrow R_{1}+R_{2}+R_{3},$ we have:
$\Delta=\left|\begin{array}{ccc}2(x+y) & 2(x+y) & 2(x+y) \\ y & x+y & x \\ x+y & x & y\end{array}\right|$
$=2(x+y)\left|\begin{array}{ccc}1 & 1 & 1 \\ y & x+y & x \\ x+y & x & y\end{array}\right|$
Applying $C_{2} \rightarrow C_{2}-C_{1}$ and $C_{3} \rightarrow C_{3}-C_{1},$ we have:
$\Delta=2(x+y)\left|\begin{array}{ccc}1 & 0 & 0 \\ y & x & x-y \\ x+y & -y & -x\end{array}\right|$
Expanding along $R_{1},$ we have:
$\Delta=2(x+y)\left[-x^{2}+y(x-y)\right]$
$=-2(x+y)\left(x^{2}+y^{2}-y x\right)$
$=-2\left(x^{3}+y^{3}\right)$
