Gaussian Elimination
The workhorse algorithm for solving linear systems. Start with [A|b], apply three simple row operations, and read off the answer.
Elementary Row Operations
Only three moves are allowed — and they never change the solution set:
- Swap two rows (Rᵢ ↔ Rᵽ)
- Scale a row by c ≠ 0 (cRᵢ → Rᵢ)
- Add a multiple of one row to another (Rᵢ + cRᵽ → Rᵢ)
Row Echelon Form
Upper-triangular staircase. Each pivot sits to the right of the one above. Zeros fill the lower-left.
Reduced Row Echelon Form
Pivots equal 1. Zeros above AND below each pivot. The solution reads off directly: x₁ = 2, x₂ = 3, x₃ = −1.
Step by Step
- 1. Write the augmented matrix [A|b]
- 2. Find the leftmost non-zero column (pivot column)
- 3. Swap rows to put a non-zero entry at the top
- 4. Eliminate entries below the pivot using row additions
- 5. Repeat for the submatrix below and to the right
- 6. Back-substitute (or continue to RREF)
Elimination in Action
System: 2x₁ + x₂ − x₃ = 8, −3x₁ − x₂ + 2x₃ = −11, −2x₁ + x₂ + 2x₃ = −3
Back Substitution
After reaching REF, work upward from the last equation:
- Last row gives x₃ directly
- Substitute x₃ into the second row to find x₂
- Substitute both into the first row to find x₁
Partial Pivoting
When the pivot entry is zero (or very small), swap rows to bring the largest-magnitude entry to the pivot position.
Key Takeaways
- Three row operations preserve the solution set
- REF: upper-triangular staircase with pivots
- RREF: pivots = 1, zeros everywhere else
- Back substitution reads the solution from REF
- Partial pivoting ensures numerical stability
M3-L3: Solution Types
Gaussian elimination reveals not just the answer — but also when NO answer exists, or when INFINITELY MANY do. Next: free variables, parametric solutions, and the connection to rank.