An algorithm for solving linear systems using elementary row operations.

Phase 1. Towards row echelon form.

1. Write an augmented matrix \(M\) for the system.
2. Move all zero rows to the bottom.
3. For each row \(R\), from top to bottom:
   1. Scale \(R\) so that its leading entry is 1.
   2. For each row \(R'\) below \(R\):
      1. Replace \(R'\) so that the column below the leading one in \(R\) is 0.

Phase 2. Towards reduced row echelon form.

4. Is the last column is a pivot column?
   • If so, abort execution, for the system is inconsistent.
5. For each row \(R\), bottom to top:
   1. For each row \(R'\) above \(R\):
      1. Replace \(R'\) so that the column above the leading one in \(R\) is 0.

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• Synonyms
• Example