Row Reduction and Echelon Forms

Introduction

• Reminder of what a system of equations is.
• Introduction to row operations and matrices.
• Focus on row reduction and echelon forms.

Types of Echelon Forms

1. Echelon Form (EF)

   • All nonzero rows are above zero rows.
   • Each leading entry of a row is in a column to the right of the leading entry of the row above it.
   • Resembles a triangular form.
   • Example:
     • Leading entry: 2 with zeros below.
     • Leading entry: 1 with zeros below.
     • Leading entry: 7 with zeros below.

2. Reduced Row Echelon Form (RREF)

   • Each leading entry in a nonzero row is 1.
   • Leading 1 is the only nonzero entry in its column.
   • Example:
     • Leading 1 in each row with zeros elsewhere.
     • Rows cascade down to the right.

Terminology

• Pivot: Leading coefficient in a row.
• Pivot Position: Where the leading 1 is in RREF.
• Pivot Column: Column containing the pivot position.

Row Reduction Algorithm

1. Initial Setup: Write as an augmented matrix.
2. Pivot Column: Begin at the leftmost non-zero column.
   • Select a nonzero entry as the pivot.
   • Interchange rows if necessary.
3. Create Zeros:
   • Use row operations to create zeros below the pivot.
   • Example:
     • Row operation to make zeros: ( -2 \times \text{Row 1} + \text{Row 2} \rightarrow \text{New Row 2} ).
4. Repeat for Remaining Rows:
   • Ensure each pivot is 1 and zeros below.
5. Final Steps:
   • Ensure pivots are 1.
   • Work from rightmost pivot upwards, creating zeros above.

Example Solution

• Transform matrix to RREF.
• Ensure all rows have leading 1s and zeros elsewhere.
• Back substitution to find solution.
• Solution: ( x_1 = 5, x_2 = 3, x_3 = -1 ).

Verification

• Substitute solutions back into original equations.
• Check each equation for correctness:
  • ( x_1 + x_2 = 8 )
  • ( 2x_1 + 2x_2 + 9x_3 = 7 )
  • ( x_2 + 5x_3 = -2 )
• Confirm that equations hold true, thus verifying the solution.