The operations used to perform row reduction are called row operations.

The operations used to perform row reduction are called row operations, of which there are three types.
Type I
Switching two rows;
Type II
Multiplying a row by a non-zero scalar ;
Type III
Adding a multiple of one row to another distinct row.

It is quite easy to see the first two types of row operations won’t change the solution set, and it can also be shown that the third doesn’t either.

The following theorem tells us about when this happens.

Now to proceed in an efficient manner, it will be convenient to represent a system in terms of its essential information. This is accomplished by means of the augmented coefficient matrix or ACM corresponding to a given system. Starting with the system appearing in (eqn:sys), its associated augmented coefficient matrix is given by This is a matrix with rows and columns. Executing a number of row operations and then computing the ACM gives the same result as first forming the ACM and performing the same set of row operations on that matrix. Because the ACM contains all of the information relevant for solving the system, and is less cumbersome to work with, we will always form the ACM first, and perform row operations on that matrix.

The simplified form in which we would like to get our matrix is referred to as reduced row echelon form. This is a term which applies to matrices in general, not just augmented coefficient matrices.

The matrix is not in row echelon form, while is in row echelon but not reduced row echelon form, and is in reduced row echelon form (the reader should check this, and understand why for each example). The main fact we will need to know is

The reduced row echelon form of a matrix is unique; for that reason we will refer to the reduced row echelon form of a matrix , and write it as . When is the ACM of a system of equations, tells us essentially everything we would like to know about the original system. The way it does this is summarized by the next result.

It is important to note that when is the ACM of a system, is the ACM of another system equivalent to the original one, which is the reduced row echelon form of the original system. In this reduced row echelon form, it is possible for an equation to consist of all zeros. If it does, we do not delete it from the system, because we want to maintain the original dimensions.

Write down the set of equations corresponding to the ACM in the previous example. Is the final equation in the system consistent or inconsistent? How does this affect the overall system?

Write down the equations corresponding to this ACM, and verify that there is a unique solution by writing it down explicitly.
Write down the four equations corresponding to this matrix and then rewrite each equation so that they express the original four variables as linear functions in only. Observe that the original variable plays the role of the independent parameter in this case, and that this is directly related to the fact that the fourth column of the ACM does not contain a leading 1. Since there is one parameter , the result is a one-parameter family of solutions.