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 ?

\begin{align}
  \begin{vmatrix}
    1 & a & a^2-bc \\ 
    1 & b & b^2-ca \\ 
    1 & c & c^2-ab 
  \end{vmatrix}
&=\begin{vmatrix}
    1 & a    & a^2-bc        \\ 
    0 & b-a  & b^2-a^2+bc-ca \\ 
    0 & c-b  & c^2-b^2+ca-ab 
  \end{vmatrix} \\
&=\begin{vmatrix}
    b-a & b^2-a^2+bc-ca \\ 
    c-b & c^2-b^2+ca-ab 
  \end{vmatrix} \\
&=\begin{vmatrix}
    b-a & (b-a)(b+a+c) \\ 
    c-b & (c-b)(c+b+a) 
  \end{vmatrix}
\end{align}
