Question

For positive numbers x, y, z the numerical value of the determinant $$ is

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

Hint:

For positive numbers x, y, z the numerical value of the determinant $$ is

Answer 1

The Correct Option is A

Step 1: Concept
Use the change of base formula: $\log_a b = \frac{\log b}{\log a}$.
Step 2: Analysis
The determinant becomes $\begin{vmatrix} \frac{\log x}{\log x} & \frac{\log y}{\log x} & \frac{\log z}{\log x}
\frac{\log x}{\log y} & \frac{\log y}{\log y} & \frac{\log z}{\log y}
\frac{\log x}{\log z} & \frac{\log y}{\log z} & \frac{\log z}{\log z} \end{vmatrix}$.
Step 3: Evaluation
Factoring out $\frac{1}{\log x}$ from $R_1$, $\frac{1}{\log y}$ from $R_2$, and $\frac{1}{\log z}$ from $R_3$, we get $\frac{1}{\log x \log y \log z} \begin{vmatrix} \log x & \log y & \log z \\ \log x & \log y & \log z \\ \log x & \log y & \log z \end{vmatrix}$.
Step 4: Conclusion
Since the rows are identical, the determinant value is 0.
Final Answer: (a)