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Properties of Determinants

Given A =|[ ...

Question

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

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Solution

Taking $2$ common from $R_{2} and C_{2}$
$B = 4 ∣ ∣ ∣ ∣ \begin{matrix} f & d & e n & l & m c & a & b \end{matrix} ∣ ∣ ∣ ∣$

$= 2 ∣ ∣ ∣ ∣ \begin{matrix} 2 f & d & e 2 n & l & m 2 c & a & b \end{matrix} ∣ ∣ ∣ ∣$

$R_{3} \leftrightarrow R_{2}, then R_{2} \leftrightarrow R_{1}$
$= 2 ∣ ∣ ∣ ∣ \begin{matrix} 2 c & a & b 2 f & d & e 2 n & l & m \end{matrix} ∣ ∣ ∣ ∣$

$C_{1} \leftrightarrow C_{2} and then C_{2} \leftrightarrow C_{3}$
$B = 2 ∣ ∣ ∣ ∣ \begin{matrix} a & b & 2 c d & e & 2 f l & m & 2 n \end{matrix} ∣ ∣ ∣ ∣ = 2 A$
$\Rightarrow \frac{B}{A} = 2$

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