### Row Echelon Form

##### Reduced row echelon form vs. General solution

Suppose the augmented matrix of a system Ax = b is simplified, by the row operations, to the following reduced row echelon form.

 [ 1 2 0 0 3 ] 0 0 1 0 4 0 0 0 1 5

This means that the simplified system is

 x1 + 2x2 = 3 x3 = 4 x4 = 5

Then it is easy to get the general solution

x1 = 3 - 2x2
x3 = 4
x4 = 5

where x2 is arbitrary. For a more general case, if the following is the reduced row echelon form of [A b],

 [ 0 1 0 a1 0 b1 0 c1 ] 0 0 1 a2 0 b2 0 c2 0 0 0 0 1 b3 0 c3 0 0 0 0 0 0 1 c4

then the solution of the system Ax = b is

 x2 = c1 - a1x4 - b1x6 x3 = c2 - a2x4 - b2x6 x5 = c3 - b3x6 x7 = c4

where x1, x4, and x6 are arbitrary.

We note that the coefficients in the general solution are explicitly given in the reduced row echelon form. Therefore the general solution of Ax = b corresponds explicitly to the reduced row echelon form of [A b]. This has the following consequences:

• Since the general solution of a system of linear equations are independent of the elimination process (see here), the reduced row echelon form of a matrix is independent of the row operations used in deriving it.
• Since the shape of the row echelon form is the same as the shape of the reduced row echelon form, the shape is also independent of the choice of the row operations.