Row-equivalent augmented matrices represent equivalent systems

**Theorem: Row Equivalence and Equivalent Linear Systems**

Two matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations. Row equivalent augmented matrices represent linear systems with identical solution sets.

**Statement:** If two augmented matrices are row equivalent, then the corresponding linear systems have exactly the same solutions.

**Elementary row operations:**
1. **Row replacement**: Add a multiple of one row to another
2. **Row scaling**: Multiply a row by a nonzero constant
3. **Row interchange**: Swap two rows

**Why it works:** Each elementary row operation corresponds to an algebraically valid manipulation of equations:
- Row replacement = adding a multiple of one equation to another (does not change solutions)
- Row scaling = multiplying both sides of an equation by a nonzero constant
- Row interchange = reordering equations

**Application:** This theorem justifies Gaussian elimination. We transform the augmented matrix `[A | b]` to a simpler row-equivalent form (echelon or reduced echelon form) and read off the solution from the simpler system, knowing it is the same as the original.Copy