The number of pivots in the reduced row echelon form of a consistent system determines whether there is one or infinitely many solutions.
Created almost 9 years ago, updated 5 months ago
ID23773
Theorem: Number of Pivots and Solution Uniqueness
For a consistent linear system, 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.
**Theorem: Number of Pivots and Solution Uniqueness**
For a consistent linear system, 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.