Did you manage 12? That's excellent. Now add the final factor of 1/2 and you are done.
I am trying to calculate the area of a triangle given the points (3,3) (-1,-1) (4,1).
I am having trouble calculating the determinant
since there is a 0 in the very middle of the last determinant, the product along the diagonal is 0. The answer key says I should get 6 but I can't find where I'm going wrong
You were on the right track with your original question, you just need a few more steps.
The product of the diagonal elements of a matrix equals the determinant of a triangular matrix. Exchanging rows 2 and 3 in your last matrix and multiplying by the requisite -1 will give you an upper triangular matrix and the product of the diagonal (and the determinant) will equal 12.
Then you must divide by 2. Not for the determinant (which is 12), but because the determinant calculates the area of a parallelogram one half of which is your triangle. These two wikipedia entries explain:
Triangular matrix - Wikipedia, the free encyclopedia
Triple product - Wikipedia, the free encyclopedia
From the beginning, the determinant can be calculated in six pieces.
-3 - 1 + 12 + 4 - 3 + 3 = -4 + 16 + 0 = 12
Your row operations were impeccable. The same six steps from that form:
0 + 0 + 0 - 0 + 12 - 0 = 12
Using expansion my minors across the second row of your reduced form:
-4*(-3-0) = 12
Using expansion my minors down the first column of your reduced form:
1*(12) = 12
I think, if you do another row operation, you will see that your triangular form was not yet accomplished. Swap rows 2 and 3 and think about it again.