Question

For positive numbers $x , y , z$, the numerical value of the determinant $\begin{vmatrix}1 & \log _{x} y & \log _{x} z \\ \log _{y} x & 1 & \log _{y} z \\ \log _{z} x & \log _{z} y & 1\end{vmatrix}$ is

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

Answer 1

Answer: (A)

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