The goal of elimination
Gaussian elimination is a way to simplify an augmented matrix by row operations until the solution becomes easy to read.
The point is not to perform random moves. Each move should make the structure clearer.
Definition
RREF
Reduced row-echelon form, or RREF, is the simplified matrix form that makes pivots and free variables easy to read.
What to look for
In a row-reduced matrix, you want to see the pivots, the free variables, and any
contradiction row such as [0 0 0 | 1].
That last kind of row tells you the system has no solution.
A simple example
Worked example
Start elimination in a small system
Take the augmented matrix
Use R_2 \leftarrow R_2 - 3R_1 to remove the first entry below the pivot:
From here you can continue until the matrix is in RREF.
Pause here and step through one elimination path. After each move, ask which entry became easier to read and why that move was chosen.
Try it here
Row-reduction stepper
The live stepper lets you compare each row operation with the matrix it produces.
Start with the augmented matrix of the system.
| 1 | 2 | -1 | 3 |
| 2 | 5 | 1 | 8 |
| 0 | 1 | 2 | 4 |
Common mistake
Common mistake
REF is not yet RREF
It is easy to stop too early. In RREF, each pivot column has zeros everywhere else, not just below the pivot.
Quick check
Quick check
Is `[0 0 0 | 1]` a harmless row?
Solution
Answer
Prerequisite link
This page depends on 2.2 Augmented matrices and row operations.