First slide

Introduction to Determinants

Question

For positive numbers x, y, and z the numerical value of the determinant $| \begin{matrix} 1 & \log_{x} y & \log_{x} z \\ \log_{y} x & 1 & \log_{y} z \\ \log_{z} x & \log_{z} y & 1 \end{matrix} |$ is:

Easy

A

0
B

1
C

$\log_{e} x y z$
D

None of these

Solution

$| \begin{matrix} 1 & \log_{x} y & \log_{x} z \\ \log_{y} x & 1 & \log_{y} z \\ \log_{z} x & \log_{z} y & 1 \end{matrix} |$
$= (1 - \log_{x} y \log_{y} z) - \log_{x} y (\log_{y} x - \log_{z} x \log_{y} z) +$ $\log_{x} z (\log_{y} x \log_{x} y - \log_{z} x)$
= $(1 - 1) - (1 - \log_{x} y \log_{y} x) + (\log_{x} z \log_{z} x - 1) = 0$ ${\sin c e \log_{x} y \log_{y} x = 1}$

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