Question

If \( a, b, c \) are positive real numbers each distinct from unity, then the value of the determinant \[ \begin{vmatrix} 1 & \log_a b & \log_a c \\ \log_b a & 1 & \log_b c \\ \log_c a & \log_c b & 1 \end{vmatrix} \] is:

• A. \( 0 \)
• B. \( 1 \)
• C. \( \log_e(abc) \)
• D. \( \log_a e \cdot \log_b e \cdot \log_c e \)

Hint:

For log determinants: Convert logs using \( \log_a b = \frac{\ln b}{\ln a} \). Try row scaling to expose symmetry. Identical or proportional rows \( \Rightarrow \) determinant = 0.

Answer 1

The Correct Option is A