Row equivalent matrices represent equivalent linear 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.