Question

Without expanding find the value of

$\left|\begin{array}{ccc}4& a& b+c\\ 4& b& c+a\\ 4& c& a+b\end{array}\right|$

Answer 1

$\left|\begin{array}{ccc}4& a& b+c\\ 4& b& c+a\\ 4& c& a+b\end{array}\right|$

$=4\left|\begin{array}{ccc}1& a+b+c& b+c\\ 1& a+b+c& c+a\\ 1& a+b+c& a+b\end{array}\right|$ [ taking $4$ common and $c_1+c_2+c_3$ ]

$=4(a+b+c)\left|\begin{array}{ccc}1& 1& b+c\\ 1& 1& c+a\\ 1& 1& a+b\end{array}\right|$

$=4(a+b+c)\times 0$

$=0$

Hence, the answer is $0.$