Row Reduction to Echelon Form is a fundamental algorithmic procedure within Linear Algebra that transforms any matrix into a canonical form characterized by leading entries in pivot positions and zero rows at the bottom. The core principle relies on elementary row operations—swapping, scaling, and adding multiples of rows—to systematically eliminate coefficients below or above pivots without altering the solution space's dimensionality properties. This method establishes the structural basis for determining matrix rank, identifying free variables, and analyzing vector subspace relationships through a standardized geometric representation known as Row Echelon Form (REF) or Reduced Row Echelon Form (RREF).
Carries an augmented matrix to row echelon form with elementary row operations and reads off the solution, defining pivots and the echelon staircase pattern.