Question

Calculate the values of the determinants:
$\left|\begin{array}{ccc}b+c& a& a\\ b& c+a& b\\ c& c& a+b\end{array}\right|$.

Answer 1

$\Delta =\left|\begin{array}{ccc}b+c& a& a\\ b& c+a& b\\ c& c& a+b\end{array}\right|R_1=R_1-R_2-R_3\Rightarrow \Delta =\left|\begin{array}{ccc}0& -2c& -2b\\ b& c+a& b\\ c& c& a+b\end{array}\right|$

Expanding along $R_1$

$\Rightarrow \Delta =0-(-2c)\left\{b\right(a+b)-bc)\}+(-2b\left)\right\{bc-c(c+a)\}\Rightarrow \Delta =2bc(a+b-c)-2bc(b-c-a)\Rightarrow \Delta =2bc(a+b-c-b+c+a)\Rightarrow \Delta =4abc$