A **linear equation** in *n* unknowns
is an equation of the form

where the coefficients and

When the number of unknowns is little, the unknowns can be denoted by .

A **system of linear equations** is a set of linear equations with the same unknowns and coefficients in the same field:

A

The triple (1,5,-4) is a solution of the system ( 2), and the triple (1,2,1) is not a solution of the system ( 2). To check this, just replace into the equations.

To this system we associate the **augmented matrix** of the system:

Erasing the last column of the augmented matrix, we get the **restricted matrix** or **coefficient matrix** of
the system:

and the restricted matrix of the system ( 2) is

In order to distinguish the restricted part of the augmented matrix from the last column, which has a different meaning, we introduce sometimes a separation line into the matrix, as displayed in the following matrix:

- 1.
- If there are rows containing only zeros, they are at the bottom of the matrix;
- 2.
- If two successive rows contain non zero entries, the second row starts with more zeros than the first one.

is in row-echelon form.

is not in row-echelon form.

- 1.
- The matrix is in row-echelon form;
- 2.
- The leftmost non zero entry (called
**leading entry**) on each row is equal to 1; - 3.
- If a column contains a leading entry, then all other entrues in this column are equal to 0.

is in row-echelon form, but is not row-reduced.

is not in row-reduced echelon form, as in the entries above the pivot in the last column are not equal to 0.