If \(x,y,z\) are positive number \(\left|\begin{array}{lll}1& \log_{x}y & \log_{x}z\\ \log_{y}x& 1 & \log_{y}z\\ \log_{z}x & \log_{z}y&1\end{array}\right|\) is equal to 

\(0\)

\(\left|\begin{array}{lll}1& \log_{x}y & \log_{x}z\\ \log_{y}x& 1 & \log_{y}z\\ \log_{z}x & \log_{z}y&1\end{array}\right|\)

$=\frac{1}{\log_{x}\log_{y}\log_{z}}\begin{vmatrix}\log_{x}&\log_{y}&\log_{z}\\\log_{x}&\log_{y}&\log_{z}\\\log_{x}&\log_{y}&\log_{z}\end{vmatrix}$

$=\frac{\log_{x}\log_{y}\log_{z}}{\log_{x}\log_{y}\log_{z}}\begin{vmatrix}1&1&1\\1&1&1\\1&1&1\end{vmatrix}$

$=0$