Question

If $x,y$ and $z$ be greater than $1$, then the value of $\left|\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 1& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 1\end{array}\right|$ is

• A. $\log x.\log y.\log z$
• B. $\log x+\log y+\log z$
• C. $0$
• D. $1-\left\{\right(\log x).(\log y).(\log z\left)\right\}$

Answer 1

The correct option is C $0$

$\left|\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 1& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 1\end{array}\right|$
Take $\frac{1}{\log x},\frac{1}{\log y},\frac{1}{\log z}$ from first, second, third rows respectively
$\frac{1}{\log x}.\frac{1}{\log y}.\frac{1}{\log z}\left|\begin{array}{ccc}\log x& \log y& \log z\\ \log x& \log y& \log z\\ \log x& \log y& \log z\end{array}\right|=0$