Question

Calculate the values of the determinants:
$\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& b+c& a& a\\ 1& b& c+a& b\\ 1& c& c& a+b\end{array}\right|$.

Answer 1

$\Delta =\left|\begin{array}{cccc}0& 1& 1& 1\\ 1& b+c& a& a\\ 1& b& c+a& b\\ 1& c& c& a+b\end{array}\right|C_4=C_4-C_3\Rightarrow \Delta =\left|\begin{array}{cccc}0& 1& 1& 0\\ 1& b+c& a& 0\\ 1& b& c+a& b-c-a\\ 1& c& c& a+b-c\end{array}\right|C_3=C_3-C_2\Rightarrow \Delta =\left|\begin{array}{cccc}0& 1& 0& 0\\ 1& b+c& a-b-c& 0\\ 1& b& c+a-b& b-c-a\\ 1& c& 0& a+b-c\end{array}\right|$

Expanding along $R_1$

$\Rightarrow \Delta =0-1\left|\begin{array}{ccc}1& a-b-c& 0\\ 1& c+a-b& b-c-a\\ 1& 0& a+b-c\end{array}\right|+0-0\Rightarrow \Delta =-\left|\begin{array}{ccc}1& a-b-c& 0\\ 1& c+a-b& b-c-a\\ 1& 0& a+b-c\end{array}\right|R_3=R_3-R_2\Rightarrow \Delta =-\left|\begin{array}{ccc}1& a-b-c& 0\\ 1& c+a-b& b-c-a\\ 0& b-a-c& 2a\end{array}\right|R_2=R_2-R_1\Rightarrow \Delta =-\left|\begin{array}{ccc}1& a-b-c& 0\\ 0& 2c& b-c-a\\ 0& b-a-c& 2a\end{array}\right|$

Expanding along $C_1$

$\Rightarrow \Delta =-\left\{1\right(4ac-{(b-a-c)}^2\}-0+0\Rightarrow \Delta ={(a+c-b)}^2-4ac$

$\Rightarrow \Delta =a^2+b^2+c^2+2ac-2bc-2ca-4ac\Rightarrow \Delta =a^2+b^2+c^2-2ac-2bc-2ca$