Gauss-Jordan elimination is a systematic procedure for transforming a matrix into reduced row-echelon form using a sequence of elementary row operations.

The basic procedure:

• First get a leading 1 in row 1 and use row 1 to get 0's below the leading 1.
• Then get a leading 1 in the next row and use that row to get 0's above and below the leading 1.
• And so on: repeat for each row in turn from first to last.

Here's an example of a complete row reduction.

Here's a summary of Gauss-Jordan elimination.

To reduce a matrix to reduced row-echelon form:

For each non-zero row in turn, from top to bottom:

1. Make sure the leading entry is not further right than any lower leading entry. (Swap rows if necessary.)
2. Make the leading entry into a leading 1. (If necessary, divide the row by the leading entry.)
3. Make the other non-zero entries from the column with the leading 1 into zeroes. (Subtract multiples of the row with that leading 1 from those other rows.)
4. Move any zero rows you produced to the bottom. (Swap them with lower rows.)

Sometimes it is more practical not to reduce a matrix all the way to reduced row-echelon form – simple row echelon form will do. A matrix in row-echelon form (without the "reduced") is almost in reduced row-echelon form, except that you don't require that the numbers above the leading 1's be zeroes. (The numbers below the leading ones must still be zeroes, since each leading one must be further to the right than those above it.)

Here's an example

The process for reducing a matrix to row echelon form is called Gaussian elimination, and is just a variation on Gauss-Jordan elimination: in step 3 above, don't bother making entries above the leading 1's into zeroes.

There is an important difference between the two forms, however: any matrix can be reduced to many different row-echelon forms, but can be reduced to just one reduced row-echelon form - the reduced row-echelon form of a matrix is unique.