**for**

**Background**

An important technique for solving a system
of linear equations is
to form the augmented matrix and
reduce to
reduced row echelon form.

**Definition (Reduced Row Echelon
Form).** A matrix is said to be in row-reduced
echelon form provided that

**(i)** In
each row that does not consist of all zero elements, the first
non-zero element in this row is a 1. (called. a "leading
1).

**(ii)** In
each column that contains a leading 1 of some row, all other elements
of this column are zeros.

**(iii)** In
any two successive rows with non-zero elements, the leading 1 of the
the lower row occurs farther to the right than the leading 1 of the
higher row.

**(iv)** If
there are any rows contains only zero elements then they are grouped
together at the bottom.

**Theorem (Reduced Row Echelon
Form).** The reduced row echelon form of a
matrix is unique.

**Definition
(Rank).** The number of nonzero rows in the
reduced row echelon form of a matrix is
called the rank of
and is denoted by .

**Theorem.** Consider
the m × n linear
system , where
is the augmented matrix.

**(i)** If then
the system has a unique solution.

**(ii)** If then
the system has an infinite number of solution.

**(iii)** If then
the system is inconsistent and has no
solution.

**Proof ****Row
Reduced Echelon Form** **Row
Reduced Echelon Form**

**Computer
Programs ****Row
Reduced Echelon Form** **Row
Reduced Echelon Form**

**Mathematica Subroutine (Complete
Gauss-Jordan Elimination).**

**Example 1.** Solve
the linear system of equations

**Solution
1.**

**Example 2.** Solve
the linear system of equations

**Solution
2.**

**Example 3.** Solve
the linear system

**Solution
3.**

**Example 4.** Solve
the linear system

**Solution
4.**

**Free Variables**

When the linear system is underdetermined, we
needed to introduce free variables in the proper location. The
following subroutine will rearrange the equations and introduce free
variables in the location they are needed. Then all that
is needed to do is find the row reduced echelon form a second
time. This is done at the end of the next example.

**Mathematica Subroutine (Under Determined
Equations).**

**Example 5.** Solve
the linear system

**Solution
5.**

**Research Experience for
Undergraduates**

**Row
Reduced Echelon Form** **Row
Reduced Echelon Form** Internet hyperlinks to
web sites and a bibliography of articles.

**Download this
Mathematica Notebook**
**Row
Reduced Echelon Form**

**Return
to Numerical Methods - Numerical Analysis**

(c) John H. Mathews 2004