1. 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.
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