Question

The value of the determinant
$\Delta =\left|\begin{array}{c}\log x\\ \log 2x\\ \log 3x\end{array}\begin{array}{c}\log y\\ \log 2y\\ \log 3y\end{array}\begin{array}{c}\log z\\ \log 2z\\ \log 3z\end{array}\right|$

• A. $0$
• B. $\log \left(xyz\right)$
• C. $\log \left(6xyz\right)$
• D. $6\log \left(xyz\right)$

Answer 1

Answer: (A)

$0$

$△=\left|\begin{array}{ccc}\log x& \log y& \log z\\ \log 2x& \log 2y& \log 2z\\ \log 3x& \log 3y& \log 3z\end{array}\right|$

$\log \left(AB\right)=\log A+\log B$- Apply for $R_2\&R_3$

$\log 2x=\log 2+\log x$

$\Rightarrow △=\left|\begin{array}{ccc}\log x& \log y& \log z\\ \log 2& \log 2& \log 2\\ \log 3& \log 3& \log 3\end{array}\right|$

$\Rightarrow △=\log 2\log 3\left|\begin{array}{ccc}\log x& \log y& \log z\\ 1& 1& 1\\ 1& 1& 1\end{array}\right|$

$\Rightarrow △=0$