Linear Algebra determinant reduction Prove, without expanding, that
\begin{vmatrix}
1 &a  &a^2-bc \\ 
1 &b  &b^2-ca \\ 
1 &c  &c^2-ab 
\end{vmatrix} vanishes.
Any hints ?

Simply add a multiple of the first column to the last
$$\begin{vmatrix} 1 & a & a^2-bc\\1&b&b^2-ac\\1&c&c^2-ab\end{vmatrix}
=\begin{vmatrix} 1 & a & a^2-bc+(1)(ab+ac+bc)\\1&b&b^2-ac+(1)(ab+ac+bc)\\1&c&c^2-ab+(1)(ab+ac+bc)\end{vmatrix}
= \begin{vmatrix} 1 & a & a(a+b+c)\\1&b&b(a+b+c)\\1&c&c(a+b+c)\end{vmatrix}
= 0$$
The final step follows because the second and third column are linearly dependent.
