Factorizing Determinants

Question

I don't know how to factorize the determinants. Please help.

1. $$ \begin{vmatrix} a+b &b+c &c+a\\ b+c &c+a &a+b\\ c+a &a+b &b+c \end{vmatrix} $$ 2. $$ \begin{vmatrix} a^2 &b^2 &c^2\\ b^2 &c^2 &a^2\\ c^2 &a^2 &b^2 \end{vmatrix} $$

Answer 1

1. Add the two last columns to the first one and then subtruct the first row from the two other rows. Now developp along the first column to find $$2(a+b+c)\left[-(a-b)^2-(b-c)(a-c)\right]$$
2. Repeat the same idea as in 1.

Answer 2

Using Saruss' rule you should easily see that the determinant of a matrix of the form $$\left(\begin{matrix}x&y&z\\y&z&x\\z&x&y\end{matrix}\right)$$ is $3xyz-x^3-y^3-z^3$.

Can you use this to solve your problems?