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# Reduced Row Echelon Form of Homogeneous Systems of Equations

5. In general, a matrix's row echelon form can vary a bit. A matrix's reduced row echelon
form is always unique. In other words, there is only one specific reduced row echelon form
matrix associated with each matrix.
(a) Consider the following homogeneous system:
2x1 ¡ x2 + x4 + 4x5 = 0
2x1 ¡ 2x2 + x3 + 4x4 ¡ 3x5 = 0
2x1 ¡ 4x2 + x3 + 6x4 + 6x5 = 0
Write this system as an augmented matrix, use your calculator and ¯nd the row echelon
form, and the reduced row echelon form.
(b) Consider the following homogeneous system:
2x1 ¡ 2x2 + x3 + 4x4 ¡ 3x5 = 0
2x1 ¡ x2 + x4 + 4x5 = 0
2x1 ¡ 4x2 + x3 + 6x4 + 6x5 = 0
Write this system as an augmented matrix. Use your calculator and ¯nd the row echelon
form and the reduced row echelon form. How is this system di®erent than the ¯rst? What
row operation would move the augmented matrix from part (a) to this one?
(c) Consider the following homogeneous system:
2x1 - 4x2 + x3 + 6x4 + 6x5 = 0
2x1 - x2 + x4 + 4x5 = 0
2x1 - 2x2 + x3 + 4x4 - 3x5 = 0
Write this system as an augmented matrix. Use your calculator and find the row echelon
form and the reduced row echelon form. How is this system different than the first? What
row operation would move the augmented matrix from part (a) to this one?
(d) Explain why the row echelon forms found in parts (b) and (c) are alternative forms of
the row echelon form matrix found in part (a).
(e) Comparing the three different row echelon forms, how are they different? What is the
same about them? What does this suggest about row echelon form of any matrix?
(f) How are the reduced row echelon forms different for the above augmented matrices?
What does this suggest about the reduced row echelon form on any matrix?
(g) While we are here, solve the homogeneous system.

#### Solution Preview

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(a) The augmented matrix of the system is:

The REF form of a matrix is not unique.
The RREF form is unique and is:

1 0 0 0 -0.25 0
0 1 0 -1 -4.5 0
0 0 1 2 -11.5 0

(b) The augmented matrix of the system is:

The REF form of a matrix is not unique.
The RREF form is unique and is:

1 0 0 0 ...

#### Solution Summary

Reduced Row Echelon Form of Homogeneous Systems of Equations is investigated.

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