RREF Step-by-Step Examples | Generated by AI
Example 1: Simple 3×3 Matrix
Starting matrix:
[ 2 4 -2 | 2 ]
[ 1 3 1 | 4 ]
[ 3 7 -1 | 8 ]
Step 1: Make first pivot = 1
Divide Row 1 by 2:
[ 1 2 -1 | 1 ]
[ 1 3 1 | 4 ]
[ 3 7 -1 | 8 ]
Step 2: Eliminate below the first pivot
- Row 2 = Row 2 - Row 1
- Row 3 = Row 3 - 3(Row 1)
[ 1 2 -1 | 1 ]
[ 0 1 2 | 3 ]
[ 0 1 2 | 5 ]
Step 3: Make second pivot = 1 (already is!)
The second pivot (in Row 2, Column 2) is already 1.
Step 4: Eliminate above and below second pivot
- Row 1 = Row 1 - 2(Row 2)
- Row 3 = Row 3 - Row 2
[ 1 0 -5 | -5 ]
[ 0 1 2 | 3 ]
[ 0 0 0 | 2 ]
Step 5: Move to third column
The third column has no pivot (it’s all zeros below), so we’re done!
Final RREF:
[ 1 0 -5 | -5 ]
[ 0 1 2 | 3 ]
[ 0 0 0 | 2 ]
Notice: Leading 1s in a staircase pattern, zeros below AND above each pivot, zeros on the left.
Example 2: 3×4 Matrix (More Zeros)
Starting matrix:
[ 1 2 3 4 | 5 ]
[ 2 4 7 8 | 12 ]
[ 1 2 5 6 | 9 ]
Step 1: First pivot is already 1
Column 1, Row 1 has a 1. Good!
Step 2: Eliminate below first pivot
- Row 2 = Row 2 - 2(Row 1)
- Row 3 = Row 3 - Row 1
[ 1 2 3 4 | 5 ]
[ 0 0 1 0 | 2 ]
[ 0 0 2 2 | 4 ]
Step 3: Move to second column
Column 2 is all zeros below the first row. Skip to column 3!
Step 4: Make third pivot = 1
Row 2 already has a 1 in Column 3. Move it to Row 2 (already there).
Step 5: Eliminate above and below third pivot
- Row 1 = Row 1 - 3(Row 2)
- Row 3 = Row 3 - 2(Row 2)
[ 1 2 0 4 | -1 ]
[ 0 0 1 0 | 2 ]
[ 0 0 0 2 | 0 ]
Step 6: Make fourth pivot = 1
Divide Row 3 by 2:
[ 1 2 0 4 | -1 ]
[ 0 0 1 0 | 2 ]
[ 0 0 0 1 | 0 ]
Step 7: Eliminate above fourth pivot
Row 1 = Row 1 - 4(Row 3):
[ 1 2 0 0 | -1 ]
[ 0 0 1 0 | 2 ]
[ 0 0 0 1 | 0 ]
Final RREF:
[ 1 2 0 0 | -1 ]
[ 0 0 1 0 | 2 ]
[ 0 0 0 1 | 0 ]
Notice: Pivots in columns 1, 3, and 4. Column 2 has no pivot (it’s a free variable). Zeros everywhere else in pivot columns.
Key Points to Remember
- Always work left to right, top to bottom
- One column at a time – finish it completely before moving on
- Each pivot becomes 1
- Eliminate BOTH above and below each pivot (that’s what makes it “reduced”)
- Once a column is done, don’t mess with it – only work below the current row for new pivots
- If a column has all zeros, skip it (no pivot there)
The systematic approach prevents you from going in circles!