The number of pivots in the reduced row echelon form of a consistent system determines whether there is one or infinitely many solutions.

**Theorem: Number of Pivots and Solution Uniqueness**

For a consistent linear system over an infinite field, such as the real or complex numbers, the number of pivots in the reduced row echelon form (RREF) determines whether the solution is unique or there are infinitely many solutions.

**Statement:** Let `A` be an `m × n` matrix. If the system `Ax = b` is consistent, then:
- If the RREF of `[A | b]` has **n pivots** (one in every variable column), the solution is **unique**.
- If the RREF has **fewer than n pivots**, there are **infinitely many solutions**.

**Explanation:** Each pivot corresponds to a basic (leading) variable that is uniquely determined. Each non-pivot column corresponds to a free variable that can take any value. The presence of even one free variable generates infinitely many solutions.

**Examples:**
- 3 variables, 3 pivots → unique solution
- 3 variables, 2 pivots → 1 free variable → infinitely many solutions (a line of solutions)
- 3 variables, 1 pivot → 2 free variables → infinitely many solutions (a plane of solutions)

**Special case:** If A is square (n × n) and invertible, it has n pivots, so `Ax = b` has exactly one solution for every b.Copy