First slide

Introduction to Determinants

Question

Let $x, y, z$ be positive and $x, y, z \neq 1$ Let $Δ = |\begin{array}{ccc} 1 & \log_{x} y & \log_{x} z \\ \log_{y} x & 1 & \log_{y} z \\ \log_{z} x & \log_{z} y & 1 \end{array}|$ then numerical value of $∆$ is

Moderate

A

-1
B

0
C

1
D

none of these

Solution

Using change of base formula
$\log_{a} b = \frac{\log b}{\log a}, a, b > 0, a, b \neq 1$
we can write
$Δ = \frac{1}{\log x \log y \log z} |\begin{array}{lll} \log x \log y \log z \\ \log x \log y \log z \\ \log x \log y \log z \end{array}| = 0$

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