Q77 / 200 · Manipal (MET) 2010 · Mathematics

For positive numbers $x, y, z$ the numerical value of the determinant $\left|\begin{array}{ccc}1 & \log _x y & \log _x z \\ \log _y x & 1 & \log _y z \\ \log _z x & \log _z y & 1\end{array}\right|$ is

  • Correct answer
    A
    0
  • B
    1
  • C
    2
  • D
    None of these

Solution

We have, $\left|\begin{array}{ccc}1 & \log _x y & \log _x z \\ \log _y x & 1 & \log _y z \\ \log _z x & \log _z y & 1\end{array}\right|$ $=\left|\begin{array}{ccc}1 & \frac{\log y}{\log x} & \frac{\log z}{\log x} \\ \frac{\log x}{\log y} & 1 & \frac{\log z}{\log y} \\ \frac{\log x}{\log z} & \frac{\log y}{\log z} & 1\end{array}\right|$ $=\frac{1}{\log x \cdot \log y \cdot \log z}\left|\begin{array}{lll}\log x & \log y & \log z \\ \log x & \log y & \log z \\ \log x & \log y & \log z\end{array}\right|=0$ $[\because$ all rows are identical]

Hence, the correct answer is option: 'A'.

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