For positive numbers x,y and z the numerical value of the determi… - Vidyadip

MCQ

For positive numbers $x,y$ and $z$ the numerical value of the determinant $\left| {\,\begin{array}{*{20}{c}}1&{{{\log }_x}y}&{{{\log }_x}z}\\{{{\log }_y}x}&1&{{{\log }_y}z}\\{{{\log }_z}x}&{{{\log }_z}y}&1\end{array}\,} \right|$is

✓
$0$
B
$1$
C
${\log _e}xyz$
D
None of these

✓

Answer

Correct option: A.

$0$

a
(a) $\left| {\,\begin{array}{*{20}{c}}1&{{{\log }_x}y}&{{{\log }_x}z}\\{{{\log }_y}x}&1&{{{\log }_y}z}\\{{{\log }_z}x}&{{{\log }_z}y}&1\end{array}\,} \right|$

= $(1 - {\log _z}y{\log _y}z) - {\log _x}y({\log _y}x - {\log _z}x{\log _y}z)$$ + {\log _x}z({\log _y}x{\log _z}y - {\log _z}x)$

= $(1 - 1)\, - (1 - {\log _x}y{\log _y}x) + ({\log _x}z{\log _z}x - 1) = 0$ $\{$ Since ${\log _x}y.{\log _y}x = 1\} $

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