# Thread: Gauss-Jordan Reduction

1. ## Gauss-Jordan Reduction

Solve the following set of homogeneous equations by Gauss-Jordan reduction of the matrix of coefficients (without the column of zeros from the right-hand side.

5x+5y-5z=0
3x+4y-7z=0
2y-8z=0
-2y-3y+6z=0

I know to do Gauss-Jordan reduction for three equations, but not four. How do I do this - I'm pretty confused

2. let's start with the matrix, which is:

[ 5 5 .5]
[ 3 4 -7]
[ 0 2 -8]
[-2 -3 6]. since it's easy, multiply the first row by 1/5. then add 2 times the (new) first row to the last row. you should have:

[1 1 .1]
[3 4 -7]
[0 2 -8]
[0 -1 8]. subtract 3 times the first row, from the 2nd, to get:

[1 1 ...1]
[0 1 -10]
[0 2 ..-8]
[0 -1 ..8]. we've effectively eliminated one variable (x) at this point. to continue, let's multiply row 3 by 1/2, and add the new row 3 to row 4:

[1 1 ...1]
[0 1 -10]
[0 1 ..-4]
[0 0 ...4]. continuing in this way, you should be able to clear out all of column two, except a 1 in row 2. if the 3rd and 4th rows are not 0, continue with column 3, with further row-operations.