Question

Show that $\left|\begin{array}{ccc}1& a& a^2\\ 1& a& b^2\\ 1& a& c^2\end{array}\right|=(a-b)(b-c)(c-a)$

Answer 1

$\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|R_2=R_1-R_2{R}_3=R_1-R_3\Rightarrow \left|\begin{array}{ccc}1& a& a^2\\ 0& a-b& a^2-b^2\\ 0& a-c& a^2-c^2\end{array}\right|\Rightarrow (a-b)(a-c)\left|\begin{array}{ccc}1& a& a^2\\ 0& 1& a+b\\ 0& 1& a+c\end{array}\right|\Rightarrow (a-b)(a-c)\times 1\left|\begin{array}{cc}1& a+b\\ 1& a+c\end{array}\right|\Rightarrow (a-b)(a-c)(a+c-a-b)\Rightarrow (a-b)(a-c)(c-b)\Rightarrow (a-b)(b-c)(c-a)$