Question

Given $A=\left|\begin{array}{ccc}a& b& 2c\\ d& e& 2f\\ l& m& 2n\end{array}\right|,B=\left|\begin{array}{ccc}f& 2d& e\\ 2n& 4l& 2m\\ c& 2a& b\end{array}\right|,$ then the value of $B/A$ is

Answer 1

Taking $2$ common from $R_2\text{and}{C}_2$
$B=4\left|\begin{array}{ccc}f& d& e\\ n& l& m\\ c& a& b\end{array}\right|$

$=2\left|\begin{array}{ccc}2f& d& e\\ 2n& l& m\\ 2c& a& b\end{array}\right|$

$R_3\leftrightarrow R_2,\text{then}{R}_2\leftrightarrow R_1$
$=2\left|\begin{array}{ccc}2c& a& b\\ 2f& d& e\\ 2n& l& m\end{array}\right|$

$C_1\leftrightarrow C_2\text{and then}{C}_2\leftrightarrow C_3$
$B=2\left|\begin{array}{ccc}a& b& 2c\\ d& e& 2f\\ l& m& 2n\end{array}\right|=2A$
$\Rightarrow \frac{B}{A}=2$