47. Augmented Matrix
When solving the linear system , the augmented matrix stacks the coefficient matrix and the right-hand side into one matrix:
The vertical bar separates the coefficients from the constants — purely visual; row operations treat it as one matrix.
Example
System:
Augmented form:
47.1. Why use it
Row-reducing to REF (via Gaussian elimination) or RREF (via Gauss–Jordan) reads off the solution directly — no need to substitute back manually if you go all the way to RREF.
47.2. Reading solutions
After row reduction:
- Unique solution:
- Infinite solutions: (free variables)
- No solution: — a row with appears
This is the Rouché–Capelli theorem expressed in terms of augmented-matrix ranks.
47.3. Connections
- Gaussian Elimination — the algorithm that operates on
- RREF — the canonical row-reduced form
- Homogeneous System — when
- Linear System Solutions