Next: Solution of a system Up: Systems of linear equations. Previous: Systems of equations and

# The Gauss-Jordan algorithm.

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.

Example 3.2.6

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