Question

The value of the determinant $\left|\begin{array}{ccc}1& a& a^2-bc\\ 1& b& b^2-ca\\ 1& c& c^2-ab\end{array}\right|$ is

Answer 1

$\left|\begin{array}{ccc}1& a& a^2-bc\\ 1& b& b^2-ca\\ 1& c& c^2-ab\end{array}\right|$

$r_1\to r_1-r_3;r_2\to r_2-r_3$

$\left|\begin{array}{ccc}0& a-c& a^2-c^2+ab-bc\\ 0& b-c& b^2-c^2+ab-ca\\ 1& c& c^2-ab\end{array}\right|=(a-c)(b-c)\left|\begin{array}{ccc}0& 1& a+b+c\\ 0& 1& a+b+c\\ 1& c& c^2-ab\end{array}\right|$

$tworowsareidentical$

$\therefore \left|\begin{array}{ccc}1& a& a^2-bc\\ 1& b& b^2-ca\\ 1& c& c^2-ab\end{array}\right|=0$