Question

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

Answer 1

$\left|\begin{array}{ccc}1& a& a^2-bc\\ 1& b& b^2-ca\\ 1& c& c^2-ab\end{array}\right|$
$=\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|+\left|\begin{array}{ccc}1& a& -bc\\ 1& b& -ca\\ 1& c& -ab\end{array}\right|$
$=\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|+\frac{1}{abc}\left|\begin{array}{ccc}a& a^2& -abc\\ b& b^2& -abc\\ c& c^2& -abc\end{array}\right|$
$=\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|-\left|\begin{array}{ccc}a& a^2& 1\\ b& b^2& 1\\ c& c^2& 1\end{array}\right|$
$C_1\leftrightarrow C_3,C_2\leftrightarrow C_3$
$=\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|-\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|=0$