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We can perform __three types__ of **elementary operations** on the rows of a matrix:

- 1.
- Interchange two rows:
- 2.
- Multiply a row by a non-zero scalar:
- 3.
- Add a row to another row:
.

A combination of the two last operations is as follows:
Add to a row a multiple of another row:
.

**Definition 3.2.1**
A matrix

*B* is

**row equivalent** to a matrix

*A* if

*B* can be obtained from

*A* by successive elementary row operations.

**Proposition 3.2.2**
If a matrix *B* can be obtained from a matrix *A* by successive elementary row operations, then the reverse process is possible too, i.e. *A* can be obtained from *B* by successive elementary row operations.

**Example 3.2.3**
On the one-hand, we have:

On the other hand, we have:

What we showed here for a single elementary operation is true for any finite
sequence of elementary operations.

**Proposition 3.2.4**
Two row equivalent matrices are the augmented matrices of two systems of linear equations having the same set of solutions.

In such a case, we say that the systems are **equivalent**.

__The Gauss-Jordan algorithm:__

**Example 3.2.5**
The last matrix is row-reduced.

**Proposition 3.2.7**
For any matrix *A*, there exists a unique row-reduced matrix *B* row-equivalent to *A*.

This means that, even if you row-reduce a matrix by two different ways (by means of elementary row operations), the row-reduced matrix you get is the same.

** Next:** Solution of a system
** Up:** Systems of linear equations.
** Previous:** Systems of equations and
*Noah Dana-Picard*

*2001-02-26*