Show the value of determinant is 0 I dont have any idea how to show the value of the following determinant is 0 without expanding  the determinants
$$
    \begin{vmatrix}
    1 & a & a^2-bc \\
    1 & b & b^2-ca \\
    1 & c & c^2-ab \\
    \end{vmatrix}
$$
Any ideas?

Subtract the first row from each of the other rows:
$$
\begin{vmatrix}
1 & a & a^2-bc \\
1 & b & b^2-ca \\
1 & c & c^2-ab \\
\end{vmatrix}
=\begin{vmatrix}
1 & a & a^2-bc \\
0 & b-a & (b-a)(b+a+c) \\
0 & c-a & (c-a)(c+a+b) \\
\end{vmatrix}\\
$$
Note that the second and third rows are both multiples of $\begin{bmatrix}0&1&a+b+c\end{bmatrix}$; therefore, they are linearly dependent.
