Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations.

The Goal

Transform the system into upper triangular form (echelon form), then use back substitution to find the solution.

Steps

1. Forward elimination — use row operations to create zeros below the pivots
2. Back substitution — solve from the last equation up

Row Operations

You can:

• Swap two rows
• Multiply a row by a nonzero scalar
• Add a multiple of one row to another

These operations don’t change the solution!

Example

$$\begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix} \xrightarrow{R_2 \to R_2 - \frac{1}{2}R_1} \begin{pmatrix} 2 & 1 \\ 0 & \frac{5}{2} \end{pmatrix}$$

The matrix is now upper triangular. Solve $y$ from row 2, then $x$ from row 1.

Pivot

The pivot is the first nonzero entry in each row. Pivots cannot be zero — if they are, swap rows.