Question

If $x,y,z$ are positive numbers, then value of the determinant $\left|\begin{array}{ccc}1& lo{g}_{x}y& lo{g}_{x}z\\ lo{g}_{y}x& 1& lo{g}_{y}z\\ lo{g}_{z}x& lo{g}_{z}y& 1\end{array}\right|$ is equal to

• A. $0$
• B. $3$
• C. $logxyz$
• D. none

Answer 1

Answer: (A)

$0$

The value of the determinant is $\left|\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 1& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 1\end{array}\right|$

$=(1-{\log }_{y}z{\log }_{z}y)-{\log }_{x}y({\log }_{y}x-{\log }_{y}z{\log }_{z}x)+{\log }_{x}z\left({\log }_{y}x{\log }_{z}y-{\log }_{z}x\right)$ $[\because log{{}_n}^m=\frac{logm}{logn}]$

$=0-1+1+1-1=0$