A) \[2{{(x+y+z)}^{2}}\]
B) \[2{{(x+y+z)}^{3}}\]
C) \[{{(x+y+z)}^{3}}\]
D) 0
Correct Answer: D
Solution :
Let \[A=\left| \begin{matrix} x+y & y+z & z+x \\ x & y & z \\ x-y & y-z & z-x \\ \end{matrix} \right|\] Applying \[{{C}_{1}}\to {{C}_{1}}+{{C}_{2}}+{{C}_{3}}\] \[=\left| \begin{matrix} 2(x+y+z) & y+z & z+x \\ x+y+z & y & z \\ 0 & y-z & z-x \\ \end{matrix} \right|\] \[=(x+y+z)\left| \begin{matrix} 2 & y+z & z+x \\ 1 & y & z \\ 0 & y-z & z-x \\ \end{matrix} \right|\] Applying \[{{R}_{2}}\to 2{{R}_{2}}-{{R}_{1}}\] \[=(x+y+z)\left| \begin{matrix} 2 & y+z & z+x \\ 0 & y-z & z-x \\ 0 & y-z & z-x \\ \end{matrix} \right|\] \[=0\] (\[\because \] Two rows are identical)
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