1. ## determinants

I'm not sure which properties that I'm supposed to play around with, any help would be appreciated!

Cheers

2. One property that is needed here is the one which says that if you add one row (or column) of a determinant to another, then the value of the determinant is unchanged.

So for example if you add each of the top two rows to the bottom row, then you get

$\displaystyle \begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix} = \begin{vmatrix}a&b&c\\b&c&a\\a+b+c&a+b+c&a+b+c\end {vmatrix}$ .

The next property that comes in handy is the one saying that if you multiply each element in a row (or column) by a constant, then the value of the determinant is multiplied by that constant. Or, to put it another way, if each element in a row (or column) has the same factor, then you can take that factor outside the determinant. Thus

$\displaystyle \begin{vmatrix}a&b&c\\b&c&a\\a+b+c&a+b+c&a+b+c\end {vmatrix} = (a+b+c)\begin{vmatrix}a&b&c\\b&c&a\\1&1&1\end{vmat rix}$ .

That gets the factor a+b+c for you. Now use similar procedures to deal with the rest of the problem.