Matrices And Determinants Ques 4
- For positive numbers $x, y$ and $z$, 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…..
(1993, 2M)
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Answer:
Correct Answer: 4.$(0)$
Solution: Let
$ \begin{aligned} \Delta & =\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| \end{aligned} $
On dividing and multiplying $R_1, R_2, R_3$ by $\log x$, $\log y, \log z$, respectively.
$ =\frac{1}{\log x \log y \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 $
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