Gaussian Elimination (a 4x4)

Date: 11/29/2001 at 23:41:32
From: Christina
Subject: Gaussian Elimination (a 4x4)

I have a problem here that is 4x4. I can do 3x3's, but I've managed to 
get myself turned around. The teacher wants us to use Gaussian 
elimination with just the matrices. If I leave everything in place and 
just break this down and solve for D, and substitute back in, I get 
(5,1,0,2). I'm assuming that is the correct answer, but its not done 
the way the teacher wants.

Here's the problem...
A  + 3B +  C + 3D = 14
4A - 2B - 3C +  D = 20
2A +  B -  C -  D = 9
A  + 2B -  C - 2D = 3

Basically after solving it should look like...
1 0 0 0 Ans
0 1 0 0 Ans
0 0 1 0 Ans

I would send what I have so far, but I've ended up with a mess of 
fractions and nowhere near what the other method gave me as an answer. 
I'm just looking for a bit of insight, maybe a map of sorts to solve.

Thanks for your help.
Christina

Date: 11/30/2001 at 01:52:22
From: Doctor Schwa
Subject: Re: Gaussian Elimination (a 4x4)

Hi Christina,

It looks as if you got it solved just fine; (5,1,0,2) certainly works.

To use Gaussian elimination, you write the same equations but without
the letters for the variables and with a vertical line in place of
the equals sign:

 1  3  1  3 | 14
 4 -2 -3  1 | 20
 2  1 -1 -1 |  9
 1  2 -1 -2 |  3

The first step is to use that 1 in the top left corner to make the 
first column into 1,0,0,0 as you desire. 
So, subtract 4 times the first row from the second row. 
Subtract 2 times the first row from the third row. 
Subtract the first row from the fourth row.

Then you'll have -14 in the (2,2) spot of your matrix.
Multiply that row times 1/-14 to get a 1 there.
Then similarly use that 1 to make the rest of its column 0.
Go on from there - and there will be lots of messy fractions -
and you'll eventually get to an answer.

I think Gaussian elimination is usually a good method for a computer 
to use, but can get pretty unpleasant for people, who can often see
shortcuts that help them solve the system more efficiently. On the
other hand, when people try to follow the shortcuts instead of working
systematically through the matrix, sometimes they end up going in 
circles.

There's a nice explanation of how to do a (slightly modified) form of 
Gaussian elimination in our Dr. Math archives:

   Solving a 6x6 System of Equations
   http://mathforum.org/library/drmath/view/55482.html 

The slight modification is that instead of just looking at the (2,2)
spot, Dr. Rob looked at the (2,3) and (2,4) spots also, and picked
the one that seemed easiest to divide by, to help keep the fractions
from getting so messy.

I hope that helps clear things up. I am afraid that to give you much 
more specific help with this problem, you'll need to send in some of 
your work with the messy fractions (not necessarily all of it, but at 
least a sampling of the first few steps you did). Feel free to write 
back with that information if you'd like more help.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/