Lesson 10: Gaussian Elimination – Row Reduction

1. What is Gaussian Elimination?

Gaussian elimination (or row reduction) is a systematic way to solve systems of linear equations by transforming the augmented matrix into row echelon form.

Goal: turn the system into an easier form where we can read off the solution (back-substitution).

Example system:

\[ \begin{cases} 2x + 3y = 8 \\ 4x - y = 2 \end{cases} \quad \rightarrow \quad \begin{pmatrix} 2 & 3 & | & 8 \\ 4 & -1 & | & 2 \end{pmatrix} \]

2. Three Elementary Row Operations

We can perform these operations without changing the solution:

1. Swap two rows
2. Multiply a row by a non-zero scalar
3. Add a multiple of one row to another row

We use these to create zeros below pivots (leading 1s or non-zeros).

3. Steps of Gaussian Elimination

1. Make the leading entry in row 1 = 1 (divide row by pivot)
2. Use row 1 to eliminate entries below pivot (make zeros)
3. Repeat for next pivot (row 2, column 2), etc.
4. Result: row echelon form (triangular shape)
5. Back-substitution to solve for variables

Reduced row echelon form goes further (1s only on pivots, zeros above too).

Summary – Lesson 10

• Gaussian elimination = row reduction to solve systems
• Three allowed row operations
• Goal: row echelon form → back-substitution
• Next: classifying solutions using rank & row echelon form