# Matrix Determinants

1. Jul 7, 2010

1. The problem statement, all variables and given/known data
Find the Determinant.

$$\left[\begin{array}{cccc}5&3&0&6\\4&6&4&12\\0&2&-3&4\\0&1&-2&2\end{array}\right]$$

2. Relevant equations

3. The attempt at a solution
I'm not sure why I cant get the determinant of this one right.

I chose to use the left most column to expand on because the two zeros make it easier

I have 2 cofactors since the other 2 would be multiplied by 0. The 4 is negative in $$C_{31}$$ because of the plus minus alternating pattern that is applied to the matrix.

$$C_{41}=5\left|\begin{array}{ccc}6&4&12\\2&-3&4\\-1&-2&2\end{array}\right|$$
$$C_{31}=-4\left|\begin{array}{ccc}3&0&6\\2&-3&4\\1&-2&2\end{array}\right|$$

$$C_{41}=5(-104)$$
$$C_{31}=-4(0)$$

If you add these you get -520, but the determinant of the 4x4 is 0. I also checked the determents of the smaller 3x3 cofactors and they are right. What am I doing wrong?

2. Jul 7, 2010

### Staff: Mentor

Check your arithmetic. I get 0 for both cofactors.

3. Jul 7, 2010

### Karmalo

The last line should be [1, -2, 2] instead of [-1, -2, 2]. That would make it 0

4. Jul 7, 2010