The value of the determinant ∣ ∣ ∣ ∣ ∣ ∣ l o g a ( x y ) l o g a ( y z ) l o g a ( z x ) l o g b ( y z ) l o g b ( z x ) l o g b ( x y ) l o g c ( z x ) l o g c ( x y ) l o g c ( y z ) ∣ ∣ ∣ ∣ ∣ ∣ is Options: A . &nbsp1 B . &nbsp-1 C . &nbsplogcxyz D . &nbspNone of these

Answer: Option D : D Applying C 1 → C 1 + C 2 + C 3 Then, ∣ ∣ ∣ ∣ ∣ ∣ 0 l o g a ( y z ) l o g a ( z x ) 0 l o g b ( z x ) l o g b ( x y ) 0 l o g c ( x y ) l o g c ( y z ) ∣ ∣ ∣ ∣ ∣ ∣ = 0

The value of the determinant ∣∣ ∣ ∣ ∣&#

Question

The value of the determinant

∣ ∣∣∣∣ ∣ \begin{matrix} l o g_{a} (\frac{x}{y}) & l o g_{a} (\frac{y}{z}) & l o g_{a} (\frac{z}{x}) l o g_{b} (\frac{y}{z}) & l o g_{b} (\frac{z}{x}) & l o g_{b} (\frac{x}{y}) l o g_{c} (\frac{z}{x}) & l o g_{c} (\frac{x}{y}) & l o g_{c} (\frac{y}{z}) \end{matrix} ∣ ∣∣∣∣ ∣

Options:

A . 1

B . -1

C . logcxyz

D . None of these

Answer: Option D
:
D
Applying

C_{1} \to C_{1} + C_{2} + C_{3}

Then,

∣ ∣∣∣∣ ∣ \begin{matrix} 0 & l o g_{a} (\frac{y}{z}) & l o g_{a} (\frac{z}{x}) 0 & l o g_{b} (\frac{z}{x}) & l o g_{b} (\frac{x}{y}) 0 & l o g_{c} (\frac{x}{y}) & l o g_{c} (\frac{y}{z}) \end{matrix} ∣ ∣∣∣∣ ∣ = 0

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