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