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Definition of (row) echelon form of a matrix

Created over 8 years ago, updated 10 days ago

Definition: The (row) echelon form of a matrix $A$ is any matrix $U$ in row echelon form such that $U$ is row-equivalent to $A$ (i.e., $U$ can be obtained from $A$ by a sequence of elementary row operations).

Key points:

  • A matrix can have many echelon forms (depending on the sequence of row operations chosen).
  • However, every matrix has exactly one reduced row echelon form (RREF).
  • The pivot positions are the same in all echelon forms of a given matrix.
  • The echelon form reveals: rank (number of pivots), consistency of the associated linear system, and which variables are free vs. basic.