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 ?

Let
$$f(a)=\begin{vmatrix}
1 &a  &a^2-bc \\ 
1 &b  &b^2-ca \\ 
1 &c  &c^2-ab 
\end{vmatrix}$$
then it's easy to see that $f$ is a polynomial on $a$ with degree at most $2$ and $f(b)=f(c)=0$ so
$$f(a)=\lambda(a-b)(a-c)$$
now, WLOG assume that $bc\ne0$, take $a=0$; we see easily that $f(0)=0$ hence $\lambda=0$.
