# Thread: Gaussian elimination and determinant

1. ## Gaussian elimination and determinant

When I do gaussian elimination on a 3x3 matrix to get a matrix in row echelon form is the bottom right entry the determinant of the original matrix?

2. No. Take a matrix that's already in row-echelon form. Since it's triangular, its determinant is the product of the diagonal entries. Now do Gaussian elimination to put it into row-echelon form (of course you do nothing in this step, but that doesn't matter). Now if your conjecture was true, the determinant would be equal to the bottom-right entry.

3. What exactly is the product of the diagonal entries? I've looked at a couple of examples I've done and the bottom right entry is the determinant.

4. Have you misunderstood my question? Is the bottom right entry of a matrix in row echelon form the determinant of the original matrix?

5. Originally Posted by Stuck Man
Have you misunderstood my question? Is the bottom right entry of a matrix in row echelon form the determinant of the original matrix?
Consider, as Tinyboss suggested, a matrix in upper triangular form:
$\begin{pmatrix} 1 & 0 & 3 \\ 0 & 4 & -2 \\ 0 & 0 & 18 \end{pmatrix}$

Is the determinant of this matrix 18?

-Dan

6. No but is it the determinant of the original matrix before it was converted?

7. Originally Posted by Stuck Man
No but is it the determinant of the original matrix before it was converted?
Tinyboss' suggestion (which I'm merely giving you a concrete example of) is that the matrix I gave you can be put into echelon form (thought it's already there) and that the determinant is not the lower right element. However I'm presuming you have a reason for saying this. So let's do this. Work out the echelon form of the following matrix and compare the lower right element with the determinant of the original:
$\begin{pmatrix} 2 & -1 \\ 3 & 2 \end{pmatrix}$

-Dan

8. I have found them to be equal but I can see now that it won't always be equal. On the other hand the bottom right entry may be the determinant multiplied by a scalar. This is what I find when there are variables.

The entry at the bottom right can be used to answer the second part of this question. I understand how to use the determinant but what is the logic here?

I do not understand the logic to the final part of the answer.

11. Originally Posted by Stuck Man