Question

For positive numbers $x,y,z$, the numerical value of the determinant $\left|\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 1& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 1\end{array}\right|$ is

• A. 0
• B. 1
• C. 2
• D. None of these

Answer 1

Answer: (A)

We have,
$\left|\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 1& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 1\end{array}\right|$
$=\left|\begin{array}{ccc}1& \frac{\log y}{\log x}& \frac{\log z}{\log x}\\ \frac{\log x}{\log y}& 1& \frac{\log z}{\log y}\\ \frac{\log x}{\log z}& \frac{\log y}{\log z}& 1\end{array}\right|$
$=\frac{1}{\log x\cdot \log y\cdot \log z}\left|\begin{array}{ccc}\log x& \log y& \log z\\ \log x& \log y& \log z\\ \log x& \log y& \log z\end{array}\right|=0$
$[\because$ all rows are identical]