Question

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

Answer 1

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

All the elements one of the rows of the determinant are $0$

$\Rightarrow \Delta =0$