\begin{vmatrix}
-2a & a+b & a+c \\
b+a & -2b & b+c \\
c+a & c+b & -2c
\end{vmatrix}

EDIT by Unknown008: Added the + signs.

| -2a a+b a+c // b+a -2b b+c // c+a c+b -2c |

I have problems with displaying the "+" sign every time I post. Is there a specific rule for writing expressions containing "+" ?

You just need to type them in manually. That is after copy/pasting, you delete the + and retype them in and it should be fine I think.

Otherwise, wikipedia has an article about that and the determinant is:

(-2a)(-2b)(-2c) + (a+b)(b+c)(c+a) + (a+c)(b+a)(c+b) - (a+c)(-2b)(c+a) - (a+b)(b+a)(-2c) - (b+c)(c+b)(-2a)

It's just like taking the diagonals:
Longest diagonal towards down-right,
Other diagonals going down-right and getting on the other side.

Then subtracting the opposite diagonals.

Sorry, it is my fault I didn't mention that I'm looking for another method (e.g. summing lines / columns or other operations) so that I can calculate it easier.

Regarding the + I typed each of them manually.

Barg, I don't know then... something to raise to the website developers, and it seems that they never took that seriously.

Otherwise, sorry, I don't know of any other method :(

OK, thank you anyway.