## Using Matrices to Solve Linear Systems

Matrices provide a systematic framework for solving linear systems through **Gaussian elimination** (row reduction). The process converts a system of equations into a matrix problem that can be solved algorithmically.

### The Method

1. **Form the augmented matrix** `[A | b]` from the system `Ax = b`
2. **Apply elementary row operations** to transform to row echelon form:
   - Swap two rows
   - Multiply a row by a nonzero scalar
   - Add a multiple of one row to another
3. **Continue to reduced row echelon form** (Gauss-Jordan elimination) for direct solution reading
4. **Back-substitute** to find variable values (if not in reduced form)

### Elementary Row Operations
These operations preserve the solution set of the system:
- **Row swap**: `Rᵢ ↔ Rⱼ`
- **Row scaling**: `Rᵢ → c·Rᵢ` (c ≠ 0)
- **Row replacement**: `Rᵢ → Rᵢ + c·Rⱼ`

### Why Matrices?
Matrix methods work for systems of any size, handle infinite and no-solution cases systematically, and are efficiently implementable on computers. They also reveal structural properties: rank, nullity, and linear independence.