# Solving a determinant to give it's factors

1. Jul 17, 2010

1. The problem statement, all variables and given/known data
I have a determinant to solve.

Fistrow is -2a, a + b, a+c
Second row is b + a, -2b, b + c
Third row is c+a, c+b, -2c

Prove that the determinant is equal to 4 (b +c) (c +a) (a + b)

2. Relevant equations

Not applicable.
3. The attempt at a solution
I have tried addig up rows and column both three at a time and two at a time. I am unable to find any common factor. This is one of my first sums, so I'msure I'm missing out on something.

Also, could you tell me how to enter a determinant at PhysicsForums?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jul 17, 2010

### scottie_000

The best way to write this is using $$\LaTeX$$, but that's best learnt another day.

What operations for finding a determinant do you know about so far?

Hint: You want to try eliminating entries in one of the rows so they become 0. What about replacing the 3rd row with (3rd+2nd+1st)?

3. Jul 17, 2010

### hunt_mat

$$\left|\begin{array}{ccc} -2a & a+b & a+c \\ b+a & -2b & b+c \\ c+a & c+b & -2c \end{array}\right|$$
Why not just expant the determinant? It's small enough to do by hand and have in mind to keep in one of the factors in the solution. It's a cheat but it will give you some idea of determinants.

4. Jul 17, 2010

i have tried adding all rows and colums.
after doing it ,i am unable to find any common factor in any row or coloumn

5. Jul 17, 2010

### Staff: Mentor

You can get fancy and perform row operations to make the determinant simpler, or you can just use brute force and expand by cofactors.

For example, expanding across the top row:
$$\left|\begin{array}{ccc} 1 & 2 & 0 \\3 & -2 & 1 \\2 & 0 & -2\end{array}\right|= 1\left|\begin{array}{cc} -2 & 1 \\0 & -2 \end{array}\right| - 2 \left|\begin{array}{cc} 3 & 1 \\2 & -2 \end{array}\right| + 0\left|\begin{array}{cc} 3 & -2 \\2 & 0 \end{array}\right|$$

Now it's a matter of evaluating three 2 x 2 determinants.