# Gaussian elimination and determinant

• May 21st 2011, 04:28 AM
Stuck Man
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?
• May 21st 2011, 05:08 AM
Tinyboss
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.
• May 21st 2011, 05:24 AM
Stuck Man
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.
• May 21st 2011, 06:12 AM
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?
• May 21st 2011, 07:20 AM
topsquark
Quote:

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
• May 21st 2011, 07:25 AM
Stuck Man
No but is it the determinant of the original matrix before it was converted?
• May 21st 2011, 08:02 AM
topsquark
Quote:

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
• May 21st 2011, 08:27 AM
Stuck Man
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.
• May 21st 2011, 09:00 AM
Stuck Man
http://img8.imageshack.us/img8/4772/capture1eq.jpg

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?
• May 21st 2011, 09:03 AM
Stuck Man
http://img819.imageshack.us/img819/3217/capture1rr.jpg

I do not understand the logic to the final part of the answer.
• May 21st 2011, 09:16 AM
topsquark
Quote:

Originally Posted by Stuck Man
http://img819.imageshack.us/img819/3217/capture1rr.jpg