Question

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

• A. 0
• B. 1
• C. \( \log_e (abc) \)
• D. \( \log_a \log_e b \log_c \)

Hint:

When dealing with determinants involving logarithms, recognize patterns of linear dependence in the rows or columns, which may lead to a determinant of zero.