The value of the determinant logax-y logay-z logaz-x logby-z logbz-x logbx-y logcz-x logcx-y logcy-z is

Applying C_1 → C_1+C_2 +C_3 Then, 0 amp; log_a(y/z) amp; log_a(z/x) 0 amp; log_b(z/x) amp; log_b(x/y) 0 amp; log_c(x/y) amp; log_c(y/z) ...

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Byju's Answer

Standard XII

Mathematics

Properties of Determinants

The value of ...

Question

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

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-1

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l o g_{c} x y z

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None of these

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Solution

The correct option is D None of these
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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Q. 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} ∣ ∣ ∣ ∣ ∣

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

Q. the value of the Determinant

∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} {log}_{a} (\frac{x}{y}) & {log}_{a} (\frac{y}{z}) & {log}_{a} (\frac{z}{x}) {log}_{b} (\frac{y}{z}) & {log}_{b} (\frac{z}{x}) & {log}_{b} (\frac{x}{y}) {log}_{c} (\frac{z}{x}) & {log}_{c} (\frac{x}{y}) & {log}_{c} (\frac{y}{z}) \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣

is equal to

Q. Find the value of z in the given determinant.

Q. The value of the determinant

∆ = |\begin{matrix} 0 & x - y & y - z \\ y - x & 0 & z - x \\ z - y & x - z & 0 \end{matrix}|

is _____________.

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The value of the determinant ∣∣ ∣ ∣ ∣∣loga(xy)loga(yz)loga(zx)logb(yz)logb(zx)logb(xy)logc(zx)logc(xy)logc(yz)∣∣ ∣ ∣ ∣∣ is

The correct option is D None of theseApplying C1→C1+C2+C3 Then, ∣∣ ∣ ∣ ∣∣0loga(yz)loga(zx)0logb(zx)logb(xy)0logc(xy)logc(yz)∣∣ ∣ ∣ ∣∣=0