Question

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

• A. $(a+b+c{)}^2$
• B. $(a+b+c{)}^3$
• C. $(a+b+c)(ab+bc+ca)$
• D. None of these

Answer 1

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

$\left|\begin{array}{ccc}a-b-c& 2a& 2a\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|\left|\begin{array}{ccc}-\sum a& 0& 2a\\ \sum a& -\sum a& 2b\\ 0& \sum a& c-a-b\end{array}\right|,\left(\begin{array}{c}{C}_1\to C_1-C_2\\ C_2\to C_2-C_3\end{array}\right)=(\sum a{)}^2\left|\begin{array}{ccc}-1& 0& 2a\\ 1& -1& 2b\\ 1& 1& c-a-b\end{array}\right|=(\sum a{)}^3,$ (on expansion)
$=(a+b+c{)}^3$