Question

$\left|\begin{array}{ccc}a-b-c& 2a& 2a\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|$ =

• A. $2{(a+b+c)}^3$
• B. ${(a-b-c)}^3$
• C. $2{(a-b-c)}^3$
• D. ${(a+b+c)}^3$

Answer 1

The correct option is D ${(a+b+c)}^3$

Performing $R_1\to R_1+R_2+R_3$
$=(a+b+c)\left|\begin{array}{ccc}1& 1& 1\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|$
$C_1\to C_1-C_2$
$C_2\to C_2-C_3$
$=(a+b+c)\left|\begin{array}{ccc}0& 0& 1\\ c+a+b& -a-b-c& 2b\\ 0& a+b+c& c-a-b\end{array}\right|$
$=(a+b+c{)}^3$