Question

Prove that
$\left|\begin{array}{ccc}bc& a& a^2\\ ca& b& b^2\\ ab& c& c^2\end{array}\right|=\left|\begin{array}{ccc}1& a^2& a^3\\ 1& b^2& b^3\\ 1& c^2& c^3\end{array}\right|$

Answer 1

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

on multiply and divide abc

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

$R_1,R_2,R_3$

$\frac{1}{abc}\left|\begin{array}{ccc}abc& a^2& a^3\\ abc& b^2& b^3\\ abc& c^2& c^3\end{array}\right|$

Common from $C_1$ to abc

$\frac{abc}{abc}\left|\begin{array}{ccc}1& a^2& a^3\\ 1& b^2& b^3\\ 1& c^2& c^3\end{array}\right|$

$\left|\begin{array}{ccc}1& a^2& a^3\\ 1& b^2& b^3\\ 1& c^2& c^3\end{array}\right|$.

R.H.S.