Question

For a positive numbers $x,y$ and $z$ the numerical value of the determinant $\left[\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 1& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 1\end{array}\right]$ is:

• A. $0$
• B. $1$
• C. ${\log }_{e}xyz$
• D. $-{\log }_{e}xyz$

Answer 1

The correct option is A $0$

$D=\left|\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 1& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 1\end{array}\right|=(1-{\log }_{z}y.{\log }_{y}z)-{\log }_{x}y({\log }_{y}x-{\log }_{z}x.{\log }_{y}z)+{\log }_{x}z({\log }_{y}x.{\log }_{z}y-{\log }_{z}x)=(1-1)-{\log }_{x}y({\log }_{y}x-{\log }_{y}x)+{\log }_{x}z({\log }_{z}x-{\log }_{z}x)=0-0+0=0$