<s>[INST] Determinant equality issue Hi I am studying for an exam tomorrow and I have a question, How do I prove that the two determinants are equal ? is there a short way ?
$2abc\left|\begin{array}{ccc}
1 & 1 & 1\\
a & b & c\\
a^{2} & b^{2} & c^{2}
\end{array}\right|$
$
\left|\begin{array}{ccc}
b+c & c+a & a+b\\
b^{2}+c^{2} & c^{2}+a^{2} & a^2+b^2\\
b^{3}+c^{3} & c^{3}+a^{3} & a^{3}+b^{3}
\end{array}\right|
$
Thanks !!!! 
 [/INST] Use 
$$abc\left|\begin{array}{ccc}
1 & 1 & 1\\
a & b & c\\
a^{2} & b^{2} & c^{2}
\end{array}\right| = \left|\begin{array}{ccc}
a & b & c\\
a^2 & b^2 & c^2\\
a^{3} & b^{3} & c^{3}
\end{array}\right| \quad\text{and}\quad
\left|\begin{array}{ccc}
0 & 1 & 1\\
1 & 0 & 1\\
1 & 1 & 0
\end{array}\right| = 2.$$
</s>