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

Created over 8 years ago, updated 10 days ago

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.