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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,zx, y, z, the value of the determinant
∣1log⁡xylog⁡xzlog⁡yx1log⁡yzlog⁡zxlog⁡zy1∣\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:
A00correct
B11
C22
DNone of These
Solution
Step 1: Apply the change of base formula Using log⁡ab=log⁡blog⁡a\log_a b = \dfrac{\log b}{\log a}, rewrite every entry:
=∣1log⁡ylog⁡xlog⁡zlog⁡xlog⁡xlog⁡y1log⁡zlog⁡ylog⁡xlog⁡zlog⁡ylog⁡z1∣= \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 1log⁡x\dfrac{1}{\log x} from row 1, 1log⁡y\dfrac{1}{\log y} from row 2, and 1log⁡z\dfrac{1}{\log z} from row 3:
=1log⁡x⋅log⁡y⋅log⁡z∣log⁡xlog⁡ylog⁡zlog⁡xlog⁡ylog⁡zlog⁡xlog⁡ylog⁡z∣= \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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