A) 0
B) 1
C) 2
D) None of these
Correct Answer: A
Solution :
We have,\[\left| \begin{matrix} 1 & {{\log }_{x}}y & {{\log }_{x}}z \\ {{\log }_{y}}x & 1 & {{\log }_{y}}z \\ {{\log }_{z}}x & {{\log }_{z}}y & 1 \\ \end{matrix} \right|\] \[=\left| \begin{matrix} 1 & \frac{\log y}{\log x} & \frac{\log z}{\log x} \\ \frac{\log x}{\log y} & 1 & \frac{\log z}{\log y} \\ \frac{\log x}{\log z} & \frac{\log y}{\log z} & 1 \\ \end{matrix} \right|\] \[=\frac{1}{\log x\cdot \log y\cdot \log z}\left| \begin{matrix} \log x & \log y & \log z \\ \log x & \log y & \log z \\ \log x & \log y & \log z \\ \end{matrix} \right|=0\] [\[\because \] all rows are identical]
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