Determinant equality issue

Question

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 !!!!

Answer 1

Use $$abc\left|\begin{array}{ccc} 1 & 1 & 1\\ a & b & c\\ a^{2} & b^{2} & c^{2} \end{array}\right| = \left|\begin{array}{ccc} a & b & c\\ a^2 & b^2 & c^2\\ a^{3} & b^{3} & c^{3} \end{array}\right| \quad\text{and}\quad \left|\begin{array}{ccc} 0 & 1 & 1\\ 1 & 0 & 1\\ 1 & 1 & 0 \end{array}\right| = 2.$$