For positive numbers x, y and z, the numerical value of the determinant <br> <img src="https://latex.codecogs.com/svg.image?\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}" /> <br> is -
Step-by-Step Solution
Key Concept: Use the property log_a b = (log b) / (log a) to rewrite the determinant elements. The determinant becomes a product of a matrix and its transpose or can be shown to have proportional rows/columns.
Let the determinant be D. Using the change of base formula, log_x y = (ln y)/(ln x), log_x z = (ln z)/(ln x), etc. The determinant becomes: <br> <img src="https://latex.codecogs.com/svg.image?\Delta = \begin{vmatrix} 1 & \frac{\ln y}{\ln x} & \frac{\ln z}{\ln x} \\ \frac{\ln x}{\ln y} & 1 & \frac{\ln z}{\ln y} \\ \frac{\ln x}{\ln z} & \frac{\ln y}{\ln z} & 1 \end{vmatrix}" /> <br> Multiply row 1 by ln x, row 2 by ln y, and row 3 by ln z: <br> <img src="https://latex.codecogs.com/svg.image?\Delta = \frac{1}{\ln x \ln y \ln z} \begin{vmatrix} \ln x & \ln y & \ln z \\ \ln x & \ln y & \ln z \\ \ln x & \ln y & \ln z \end{vmatrix}" /> <br> Since all rows are identical, the determinant is 0.
Correct Answer: (A)