Matrices & Determinants
Properties of determinants
Grade Class 12
Question:
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}" /> is -
(A) 0
(B) log xyz
(C) log(x + y + z)
(D) logx logy logz
Step-by-Step Solution
Key Concept: Use the property of logarithms log_a b = (log b) / (log a). The determinant becomes a product of terms that cancel out to zero.
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?\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 two rows are identical, the value of the determinant is 0.
Correct Answer: 1