[SOLVED]Gaussian Elimination?

urahonky

See I have to learn to do it on an exam tonight. But the problem is I'm getting conflicting answers on how to do them. Online it tells me to convert the matrix to an upper triangle using row elimination. Then when you get the final answer you put that back into the equations above it to find the answer. That seems easy enough.

But in my book it has a different method which confuses me.

Basically it says: The first equation is the pivot equation, get the pivot coefficient. Multiply the pivot equation by m21 (which is the first term of the second equation divided by the pivot coefficient) and subtract it from the second equation. Then you do this for the next two equations. Then you use the 2nd equation as the pivot, and find m32, and go down as you did before.

I've never actually used this technique to get the upper triangle. Is it something I should look into? I think (maybe) the row substitution seems easier...

Kakodaimonos

There's a few variations on Guass-Jordan elimination. It comes down to whether you want just simple row echelon form or reduced-row echelon form in the matrix.

Reduced Row Echelon Form

urahonky

Hmm I think I may have figured out why the book and online are different. The book has it set up as:

a11 a12 a13 a14  x1  =  b1
a21 a22 a23 a24  x2  =  b2
a31 a32 a33 a34  x3  =  b3
a41 a42 a43 a44  x4  =  b4

But the online ones are already in augmented form. Still... Is there a difference?

urahonky

Kakodaimonos wrote: »

There's a few variations on Guass-Jordan elimination. It comes down to whether you want just simple row echelon form or reduced-row echelon form in the matrix.

Reduced Row Echelon Form

Hmm. Now do you end up with the same answer? Echelon form gives me what x1,x2,x3,etc is really easily right? My study guide says: Learn to use the backward substitution in Gaussian Elimination. So I assume that I shouldn't use the echelon form?

Kakodaimonos

No. Augmented form [A | C] is simply a notational simplification. It makes it easier to do your matrix manipulations, but there's not a difference in how the algorithm works.

Reduced row-echelon form is simply solving explicitly for your solution vector by removing the y & z coefficients from the x row, etc.

urahonky

Sweet. Thanks Kako. I appreciate it!

e: I was never really good at Matrix Algebra.