For positive numbers x- y and z- the numerical value of the determinant Neginvmatrix 1 -log - x y -log - x z log - y x - 1 -log - y z log - z x -log - z y - 1lend vmatrix isA- log x logy logzB- log x-log y-log zC- 1D- 0

The correct option is D 0Let Δ = 1 amp; log_x y amp; log_x z log_y x amp; 1 amp; log_y z log_z x amp; log_z y amp; 1 = 1 amp; log y/l ...

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Standard XII

Mathematics

Properties of Determinants

For positive ...

Question

For positive numbers x, y and z, the numerical value of the determinant $∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & l o g_{x} y & l o g_{x} z l o g_{y} x & 1 & l o g_{y} z l o g_{z} x & l o g_{z} y & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$ is ___

l o g x l o g y l o g z

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l o g x + l o g y + l o g z

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0.0

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Solution

The correct option is C 0.0
Let $Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & l o g_{x} y & l o g_{x} z l o g_{y} x & 1 & l o g_{y} z l o g_{z} x & l o g_{z} y & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & \frac{l o g y}{l o g x} & \frac{l o g z}{l o g x} \frac{l o g x}{l o g y} & 1 & \frac{l o g z}{l o g y} \frac{l o g x}{l o g z} & \frac{l o g y}{l o g z} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣$
On dividing and multiplying $R_{1}, R_{2}, R_{3}$ by $l o g x, l o g y, l o g z$ respectively.
$= \frac{1}{l o g x l o g y l o g z} ∣ ∣ ∣ ∣ \begin{matrix} l o g x & l o g y & l o g z l o g x & l o g y & l o g z l o g x & l o g y & l o g z \end{matrix} ∣ ∣ ∣ ∣ = 0$

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For positive numbers x, y and z, the numerical value of the determinant ∣∣ ∣ ∣∣1logxylogxzlogyx1logyzlogzxlogzy1∣∣ ∣ ∣∣ is ___

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