Question

For 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. $\log xyz$
• C. $\log (x+y+z)$
• D. $\log x\log y\log z$

Answer 1

Answer: (A)

$0$

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

Applying $R_1\to \log x{R}_1,R_2\to \log y{R}_2,R_3\to \log z{R}_3$

We get $\frac{1}{\log x\log y\log z}\left|\begin{array}{ccc}\log x& \log y& \log z\\ \log x& \log y& \log z\\ \log x& \log y& \log z\end{array}\right|=0$

Hence, option A.