For positive numbers x, y and z, the numerical value of the determinant <br> <img src="https://latex.codecogs.com/svg.latex?\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}" alt="determinant"> 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 terms that simplify to 1, resulting in a matrix with identical rows or columns, or a skew-symmetric structure leading to 0.
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> D = <br> <img src="https://latex.codecogs.com/svg.latex?\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}" alt="determinant_expanded"> <br> Multiply row 1 by ln x, row 2 by ln y, and row 3 by ln z: <br> D = (1 / (ln x ln y ln z)) * <br> <img src="https://latex.codecogs.com/svg.latex?\begin{vmatrix} \ln x & \ln y & \ln z \\ \ln x & \ln y & \ln z \\ \ln x & \ln y & \ln z \end{vmatrix}" alt="determinant_simplified"> <br> Since all rows are identical, the value of the determinant is 0.
Correct Answer: A