The value of the determinant loga x-y loga y-z loga z-x logb y -z logb z -x logb x-y logc z-x logc x - y logc y-z is

The correct option is D none of these log_a (x/y) #160; amp; log_a (y/z) #160; amp; log_a (z/x) log_b (y /z) #160; amp; log_b (z /x) #1 ...

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Byju's Answer

Standard XII

Chemistry

Fruendlich Isotherm

The value of ...

Question

The value of the determinant $∣ ∣ ∣ ∣ ∣ \begin{matrix} l o g_{a} (x / y) & l o g_{a} (y / z) & l o g_{a} (z / x) l o g_{b} (y / z) & l o g_{b} (z / x) & l o g_{b} (x / y) l o g_{c} (z / x) & l o g_{c} (x / y) & l o g_{c} (y / z) \end{matrix} ∣ ∣ ∣ ∣ ∣$ is

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-1

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log

_{a}

xyz

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none of these

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Solution

The correct option is D none of these
$∣ ∣ ∣ ∣ ∣ \begin{matrix} l o g_{a} (x / y) & l o g_{a} (y / z) & l o g_{a} (z / x) l o g_{b} (y / z) & l o g_{b} (z / x) & l o g_{b} (x / y) l o g_{c} (z / x) & l o g_{c} (x / y) & l o g_{c} (y / z) \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \frac{l o g_{e} (x / y)}{{log}_{e} a} & \frac{l o g_{e} (y / z)}{{log}_{e} a} & \frac{l o g_{e} (z / x)}{{log}_{e} a} \frac{l o g_{e} (y / z)}{{log}_{e} b} & \frac{l o g_{e} (z / x)}{{log}_{e} b} & \frac{l o g_{e} (z / x)}{{log}_{e} b} \frac{l o g_{e} (x / y)}{{log}_{e} c} & \frac{l o g_{e} (x / y)}{{log}_{e} c} & \frac{l o g_{e} (y / z)}{{log}_{e} c} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣$
$= \frac{1}{{log}_{e} a {log}_{e} b {log}_{e} c} ∣ ∣ ∣ ∣ ∣ \begin{matrix} l o g_{e} (x / y) & l o g_{e} (y / z) & l o g_{e} (z / x) l o g_{e} (y / z) & l o g_{e} (z / x) & l o g_{e} (x / y) l o g_{e} (z / x) & l o g_{e} (x / y) & l o g_{e} (y / z) \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= \frac{1}{{log}_{e} a {log}_{e} b {log}_{e} c} ∣ ∣ ∣ ∣ \begin{matrix} l o g_{e} x - l o g_{e} y & l o g_{e} y - l o g_{e} z & l o g_{e} z - l o g_{e} x l o g_{e} y - l o g_{e} z & l o g_{e} z - l o g_{e} x & l o g_{e} x - l o g_{e} y l o g_{e} z - l o g_{e} x & l o g_{e} x - l o g_{e} y & l o g_{e} y - l o g_{e} z \end{matrix} ∣ ∣ ∣ ∣$
$C_{1} \to C_{1} + C_{2} + C_{3}$
$= \frac{1}{{log}_{e} a {log}_{e} b {log}_{e} c} ∣ ∣ ∣ ∣ \begin{matrix} 0 & l o g_{e} y - l o g_{e} z & l o g_{e} z - l o g_{e} x 0 & l o g_{e} z - l o g_{e} x & l o g_{e} x - l o g_{e} y 0 & l o g_{e} x - l o g_{e} y & l o g_{e} y - l o g_{e} z \end{matrix} ∣ ∣ ∣ ∣$
$= 0$

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The value of the determinant ∣∣ ∣ ∣∣loga(x/y)loga(y/z)loga(z/x)logb(y/z)logb(z/x)logb(x/y)logc(z/x)logc(x/y)logc(y/z)∣∣ ∣ ∣∣ is

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