The input of the algorithm is an m × n matrix (not necessarily square!), which is typically an augmented matrix of a linear system, however the algorithm works for any matrix with numerical entries.
Start with i = 1, j = 1.
If aij = 0 swap the i-th row with some other row below to guarantee that aij 6= 0. The non-zero entry in the (i, j)-position is called a pivot. If all entries in the column are zero, increase j by 1.
Divide the i-th row by aij to make the pivot entry = 1.
Eliminate all other entries in the j-th column by subtracting suitable multiples of the i-th row from the other rows.
Increase i by 1 and j by 1 to choose the new pivot element. Return to Step 1.
The algorithm stops after we process the last row or the last column of the matrix.
The output of the Gauss-Jordan algorithm is the matrix in reduced row-echelon form.
Reduced row-echelon form
A matrix is in reduced row-echelon form (RREF) if it satisfies all of the following conditions.
If a row has nonzero entries, then the first non-zero entry is 1 called the leading 1 in this row.
If a column contains a leading one then all other entries in that column are zero.
If a row contains a leading one the each row above contains a leading one further to the left.
The last point implies that if a matrix in rref has any zero rows they must appear as the last rows of the matrix.
