Row Echelon Form (REF)

• Definition:
  A matrix is in Row Echelon Form (REF) if it satisfies the following three conditions:

  1. All non-zero rows are above any zero rows.
     (If there are any rows consisting entirely of zeros, they must be at the bottom of the matrix.)
  2. The leading entry (the first non-zero element from the left) of each non-zero row is in a column to the right of the leading entry of the row above it.
     (This creates a "stair-step" pattern.)
  3. All entries in a column below a leading entry are zero.
     (This is a direct consequence of condition 2, ensuring the "steps" are clear.)

• Characteristics:

  • Not Unique: A given matrix can have multiple different Row Echelon Forms, depending on the sequence of elementary row operations performed.
  • Determining Rank: The number of non-zero rows (which is equivalent to the number of leading entries/pivots) in an REF matrix gives the rank of the original matrix.
  • For Augmented Matrix [A | b]:
    • Used as an intermediate step in Gaussian elimination to simplify the system.
    • Helps to quickly identify if a system is inconsistent (no solution). This occurs if an REF of [A | b] contains a row of the form [0 0 ... 0 | c], where c is a non-zero constant. Such a row implies 0 = c, which is a contradiction.
    • Allows for back-substitution to find solutions, especially in the context of Ax=b.

• Example:
  The following is a matrix in Row Echelon Form:

  $$\left[\begin{array}{cccc}\mathbf{1}& 2& 3& 4\\ 0& \mathbf{5}& 6& 7\\ 0& 0& \mathbf{8}& 9\\ 0& 0& 0& 0\end{array}\right]$$

  The leading entries (pivots) are marked in bold.

Reduced Row Echelon Form (RREF)

• Definition:
  A matrix is in Reduced Row Echelon Form (RREF) if it satisfies all the conditions for Row Echelon Form (REF), and additionally meets these two stricter conditions:

  1. Each leading entry (pivot) must be 1.
     (All pivots are "normalized" to one.)
  2. Each column containing a leading entry (pivot) must have zeros everywhere else.
     (Not only below the pivot but also above it, all other entries in that column must be zero.)

• Characteristics:

  • Uniqueness: The Reduced Row Echelon Form of any given matrix is unique. Regardless of the sequence of valid elementary row operations performed, the final RREF will always be the same.
  • Direct Solutions: For a linear system Ax = b, transforming its augmented matrix [A | b] into RREF allows for direct reading of the general solution. The pivot variables can be immediately expressed in terms of the free variables and constants.
  • Identification of Variables: RREF makes it straightforward to identify pivot variables (corresponding to pivot columns) and free variables (corresponding to non-pivot columns).
  • Null Space Basis: RREF is the critical step for finding the basis vectors for the null space of a matrix, as it directly gives the relationships between pivot and free variables for the homogeneous system Ax = 0.

• Example (from your previous notes):
  The matrix $A=\left[\begin{array}{cccc}1& 0& 8& -4\\ 0& 1& 2& 12\end{array}\right]$ is already in Reduced Row Echelon Form:

  $$\left[\begin{array}{cccc}\mathbf{1}& 0& 8& -4\\ 0& \mathbf{1}& 2& 12\end{array}\right]$$

  The leading entries (pivots) are marked in bold.

  • All pivots are 1.
  • In the columns containing pivots (column 1 and column 2), all other entries are 0.

  Here is another, more comprehensive example of an RREF matrix (which could be an augmented matrix [A|b]):

  $$\left[\begin{array}{cccccc}\mathbf{1}& 0& 0& 5& |& 10\\ 0& \mathbf{1}& 0& -2& |& 7\\ 0& 0& \mathbf{1}& 3& |& 0\end{array}\right]$$

  From this RREF, you can directly read the solutions (e.g., $x_1+5x_4=10$, $x_2-2x_4=7$, $x_3+3x_4=0$).