Gaussian Elimination/Gauss-Jordan Elimination Example by Erik Gomez

Find the Reduced Row-Echelon Form of Matrix A.

A= 2 4 6 8
3 6 9 12
4 3 2 1

First, we identify any nonzero number in the first row and the first column and use it as a pivot.
We choose entry A(1,1) which is a 2. We want to turn this 2 into a 1, so we multiply the first row by 1/2.
This gives us:

1 2 3 4
3 6 9 12
4 3 2 1

Now, we want every entry under our pivot to be a zero.
We deal first with the entry A(2,1) which is a 3. We multiply the second row by -1/3.
This gives us:

1 2 3 4
-1 -2 -3 -4
4 3 2 1

We now add the first row to the second row.
This gives us:

1 2 3 4
0 0 0 0
4 3 2 1

Now, we deal with entry A(3,1) which is a 4. We multiply the third row by -1/4.
This gives us:

1 2 3 4
0 0 0 0
-1 -3/4 -1/2 -1/4

We now add the first row to the third row.
This gives us:

1 2 3 4
0 0 0 0
0 5/4 5/2 15/4

We now identify yet another pivot but this time in the second row and the second column. Since our second row contains only zeros, we permute the second and third rows.
This gives us:

1 2 3 4
0 5/4 5/2 15/4
0 0 0 0

Now we choose the entry A(2,2), which is 5/4, as our second pivot. We want to turn 5/4 into a 1, so we multiply the second row by its reciprocal 4/5.
This gives us the Row-Echelon Form of Matrix A:

1 2 3 4
0 1 2 3
0 0 0 0 Row-Echelon Form

We then go one extra step to get the Reduced Row-Echelon Form of Matrix A.
We need all the numbers above our pivots to be zeros. If we look at the Row-Echelon Form of Matrix A, it has only one pivot with numbers above it. This means that we deal with this pivot only.
We multiply the second row by -2 and add it to the first row.
This gives us the Reduced Row-Echelon Form of Matrix A:

1 0 -1 -2
0 1 2 3
0 0 0 0 Reduced Row-Echelon Form

Proof!
Plug the matrix into Matlab/Octave and see. Copy and paste the commands.
Commands:
A=[2 4 6 8;3 6 9 12;4 3 2 1] <Enter>
rref(A) <Enter>

Remember!
A matrix has numerous Row-Echelon Forms depending on the row operations we perform and their order; however, only one Reduced Row-Echelon Form exists for every given matrix.

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