Linear systems and echelon matrices

Linear Systems and Echelon Matrices

When a linear system is represented by an augmented matrix in echelon form, the structure of the system becomes transparent. Echelon form reveals key properties about the solution set without further computation.

Row Echelon Form (REF)

A matrix is in REF if:

1. All nonzero rows are above any zero rows
2. Each leading entry (pivot) is to the right of the pivot in the row above
3. All entries below each pivot are zero

Reduced Row Echelon Form (RREF)

A matrix is in RREF if it is in REF and additionally:
4. Every pivot equals 1
5. Each pivot is the only nonzero entry in its column

What Echelon Form Reveals

• Pivot positions identify basic (leading) variables
• Non-pivot columns identify free variables
• Inconsistency check: If the last column of the augmented matrix contains a pivot, the system is inconsistent (has a row [0 0 ... 0 | c] with c ≠ 0)
• Uniqueness: For a consistent system, if every column except the last has a pivot, the solution is unique; otherwise there are infinitely many solutions