Systems of Linear Equations

Definition

A matrix is said to be in reduced row echelon form if it has the
fol lowing properties .

1. If a row does not consist entirely of zeros, then the first nonzero
number in the row is a 1, called a leading 1.

2. If there are any rows that consist entirely of zeros, then they are
grouped together at the bottom of the matrix.

3. In any two successive rows that do not consist entirely of zeros, the
leading 1 in the lower row occurs farther to the right than the
leading 1 in the higher row.

4. Each column that contains a leading 1 has zeros everywhere else.
A matrix that has the first three properties is said to be in row echelon
form.

Example

Which of the following matrices are in reduced row echelon form?

Which of the following matrices are in row echelon form?

Suppose that the augmented matrix for a linear system in the unknowns
and has been reduced by elementary row operations to

This matrix is in reduced row echelon form and corresponds to the
equations

Thus, the system has a unique solution.

Example (4)

Suppose that the augmented matrix for a linear system in the unknowns
x; y, and z has been reduced by elementary row ope rations to the given
reduced row echelon form. Solve the system .

Gauss- Jordan elimination is a procedure, using elementary row
operations, for reducing a matrix to reduced row echelon form. The
algorithm consists of two parts

a forward phase in which zeros introduced below the leading 1\'s.
a backward phase in which zeros are introduced above the leading
1\'s.

If only the forward phase is used, then the procedure produces a row
echelon form and is called Gaussian elimination.

Some Facts about Echelon Forms:

1. Every matrix has a unique reduced row echelon form.

2. Row echelon forms are not unique.

Theorem (2.2.1)

A homogeneous linear system has only the trivial (zero) solution or it has
infinitely many solutions; there are no other possibilities.

Theorem (2.2.2)

Dimension Theorem for Homogeneous Systems
If a homogeneous linear system has n unknowns, and if the reduce row
echelon form of its augmented matrix has r nonzero rows, then the
system has n - r free variables .

Theorem (2.2.3)

A homogeneous linear system with more unknowns than equations has
infinitely
many solutions.

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