Question

Evaluate the following determinants without expansion as far as possible.
$\left|\begin{array}{ccc}1& a& b+c\\ 1& b& c+a\\ 1& c& a+b\end{array}\right|$

Answer 1

$\left|\begin{array}{ccc}1& a& b+c\\ 1& b& c+a\\ 1& c& a+b\end{array}\right|$

$C_2\to C_2+C_3$

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

As when any two rows or columns are same then the determinant will be 0