The value of $\left| \begin{matrix} 1 & \log_{x}y & \log_{x}z \ \log_{y}x & 1 & \log_{y}z \ \ \log_{z}x & \log_{z}y & 1 \end{matrix} ight|$ is equal to

Question

The value of $\left| \begin{matrix} 1 & \log_{x}y & \log_{x}z \ \log_{y}x & 1 & \log_{y}z \ \ \log_{z}x & \log_{z}y & 1 \end{matrix} ight|$ is equal to

Dr. Abubacker · Accepted Answer

(a)
$\left| \begin{matrix}
1 & \log_{x}y & \log_{x}z \
\log_{y}x & 1 & \log_{y}z \
\log_{z}x & \log_{z}y & 1
\end{matrix} ight|$
$= 1(1 - \log_{y}{z\log_{z}{y) - \log_{x}{y(\log_{y}x - \log_{z}x\log_{y}{z)}}}}$
$+ \log_{x}z\ (\log_{z}{y\ }\log_{y}x - \log{\ _{z}x})$
$= (1 - \log_{y}{y) - \log_{x}{y\ (\log_{y}{x - \log_{y}{x)}}}}$
$+ \log_{x}{z(\log_{z}{x - \log_{z}{x)}}}$
$= (1 - 1) - 0 + 0 = 0$

Q. The value of $\left| \begin{matrix} 1 & \log_{x}y & \log_{x}z \\ \log_{y}x & 1 & \log_{y}z \\ \ \log_{z}x & \log_{z}y & 1 \end{matrix} \right|$ is equal to

Explanation

Reviewed by Dr. Abubacker

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