Question

The value of $\left|\begin{array}{ccc}a-b-c& 2a& 2a\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|$ is

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

Answer 1

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

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

Applying $R_1\to R_1+R_2+R_3$

$\Delta =\left|\begin{array}{ccc}a+b+c& a+b+c& a+b+c\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|\Delta =(a+b+c)\left|\begin{array}{ccc}1& 1& 1\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|$

Applying $C_2\to C_2-C_1;C_3\to C_3-C_1$

$\Delta =(a+b+c)\left|\begin{array}{ccc}1& 0& 0\\ 2b& -b-c-a& 0\\ 2c& 0& -c-a-b\end{array}\right|={(a+b+c)}^3$