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Matrices & Determinants: Positive Numbers Value Determinant Vmatrix Vmatrix

JEE Maths question with a full step-by-step solution.

Question

For positive numbers

x, y, z

, the value of the determinant

\begin{vmatrix} 1 & \log_x y & \log_x z \\ \log_y x & 1 & \log_y z \\ \log_z x & \log_z y & 1 \end{vmatrix}

is:

0

correct

1

2

DNone of These

Solution

Step 1: Apply the change of base formula Using

\log_a b = \dfrac{\log b}{\log a}

, rewrite every entry:

= \begin{vmatrix} 1 & \dfrac{\log y}{\log x} & \dfrac{\log z}{\log x} \\[6pt] \dfrac{\log x}{\log y} & 1 & \dfrac{\log z}{\log y} \\[6pt] \dfrac{\log x}{\log z} & \dfrac{\log y}{\log z} & 1 \end{vmatrix}

Step 2: Factor out from each row Factor

\dfrac{1}{\log x}

from row 1,

\dfrac{1}{\log y}

from row 2, and

\dfrac{1}{\log z}

from row 3:

= \frac{1}{\log x \cdot \log y \cdot \log z} \begin{vmatrix} \log x & \log y & \log z \\ \log x & \log y & \log z \\ \log x & \log y & \log z \end{vmatrix}

Step 3: Identify identical rows All three rows are identical, so the determinant equals zero. Answer: (1)

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