A) 9
B) \[-9\]
C) 0
D) \[-1\]
E) 1
Correct Answer: B
Solution :
\[\left| \begin{matrix} x & 3 & 6 \\ 3 & 6 & x \\ 6 & x & 3 \\ \end{matrix} \right|=\left| \begin{matrix} 2 & x & 7 \\ x & 7 & 2 \\ 7 & 2 & x \\ \end{matrix} \right|=\left| \begin{matrix} 4 & 5 & x \\ 5 & x & 4 \\ x & 4 & 5 \\ \end{matrix} \right|=0\] \[\Rightarrow \] \[\left| \begin{matrix} x+9 & x+9 & x+9 \\ 3 & 6 & x \\ 6 & x & 3 \\ \end{matrix} \right|\] \[\left| \begin{matrix} 9+x & 9+x & 9+x \\ x & 7 & 2 \\ 7 & 2 & x \\ \end{matrix} \right|\] \[=\left| \begin{matrix} 9+x & 9+x & 9+x \\ 5 & x & 2 \\ 7 & 2 & x \\ \end{matrix} \right|=0\] \[\Rightarrow \] \[(x+9)\left| \begin{matrix} 1 & 0 & 0 \\ 3 & 3 & x-3 \\ 6 & x-6 & -3 \\ \end{matrix} \right|\] \[=(9+x)\left| \begin{matrix} 1 & 0 & 0 \\ x & 7-x & 2-x \\ 7 & -5 & x-7 \\ \end{matrix} \right|\] \[=(9+x)\left| \begin{matrix} 1 & 0 & 0 \\ 5 & x-5 & 4 \\ x & 4-x & 5-x \\ \end{matrix} \right|=0\] \[\Rightarrow \] \[x+9=0\] \[\Rightarrow \] \[x=-9\]
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