Question

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

• A. $logx.logy.logz$
• B. $logx+logy+logz$
• C. $0$
• D. $1-\left\{\left(logx\right).\left(logy\right).\left(logz\right)\right\}$

Answer 1

Answer: (C)

$\left|\begin{array}{ccc}\frac{logx}{logx}& \frac{logy}{logx}& \frac{logz}{logx}\\ \frac{logx}{logy}& \frac{logy}{logy}& \frac{logz}{logy}\\ \frac{logx}{logz}& \frac{logy}{logz}& \frac{logz}{logz}\end{array}\right|$
Taking $\frac{1}{logx},\frac{1}{logy},\frac{1}{logz}$ common from $R_1,R_2,R_3$ all rows are identical. So $\Delta =0$