Question

Prove the following :
$\left|\begin{array}{ccc}a+b+2c& a& b\\ c& b+c+2a& b\\ c& a& c+a+2b\end{array}\right|=2{(a+b+c)}^3.$

Answer 1

Apply $C_1-C_2$ and $C_2-C_3$ and take $(a+b+c)$ common from each of $C_1$ and $C_2$.
$\therefore \Delta ={(a+b+c)}^2\left|\begin{array}{ccc}-1& 0& 2a\\ 1& -1& 2b\\ 0& 1& c-a-b\end{array}\right|$
Apply $R_1+R_2+R_3$
$={(a+b+c)}^2\left|\begin{array}{ccc}0& 0& a+b+c\\ 1& -1& 2b\\ 0& 1& c-a-b\end{array}\right|$
Expand with $1st$ row
$\Delta ={(a+b+c)}^3\left|\begin{array}{cc}1& -1\\ 0& 1\end{array}\right|={(a+b+c)}^3$.