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Echelon Forms & RREF

Echelon Forms & RREF

Chara Chara
September 29, 2025
3 min read
linear-algebra lecture-2
index

Echelon Form

Definition (Row Echelon Form)

A matrix is in row echelon form (REF) if:

  1. Each leading entry (first nonzero in row, called a pivot) is to the right of the pivot in the row above.
  2. All zero rows are at the bottom.

Example

[$∗∗∗∗0$∗∗∗000$∗0000000000]\begin{bmatrix} \$ & * & * & * & * \\ 0 & \$ & * & * & * \\ 0 & 0 & 0 & \$ & * \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}
  • $ = pivot (nonzero)
  • ∗* = any entry

Reduced Row Echelon Form (RREF)

Definition (Reduced Row Echelon Form)

A matrix is in RREF if:

  1. It is in echelon form.
  2. All pivots are 11 (scaling).
  3. Each pivot is the only nonzero in its column.

Example

[10∗0∗01∗0∗0001∗0000000000]\begin{bmatrix} 1 & 0 & * & 0 & * \\ 0 & 1 & * & 0 & * \\ 0 & 0 & 0 & 1 & * \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}

Example: REF but not RREF

[01240π0003002000001700000000]\begin{bmatrix} 0 & 1 & 2 & 4 & 0 & \pi & 0 \\ 0 & 0 & 3 & 0 & 0 & \sqrt{2} & 0 \\ 0 & 0 & 0 & 0 & 1 & 7 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}
  • Is in echelon form.
  • Not in reduced echelon form.

Algorithms

Getting to Echelon Form

  1. From the left, pick first nonzero column → pivot column.
  2. Swap rows so top entry is nonzero → pivot.
  3. Use pivot to kill entries below (replacement).
  4. Move to submatrix southeast of pivot.
  5. Repeat until in echelon form.

Getting to RREF

  1. Scale rows so each pivot = 11

    [p1∗∗∗0p2∗∗00p3∗0000]    →    [1∗/p1∗/p1∗/p101∗/p2∗/p2001∗/p30000]\begin{bmatrix} p_1 & * & * & * \\ 0 & p_2 & * & * \\ 0 & 0 & p_3 & * \\ 0 & 0 & 0 & 0 \end{bmatrix} \;\;\to\;\; \begin{bmatrix} 1 & */p_1 & */p_1 & */p_1 \\ 0 & 1 & */p_2 & */p_2 \\ 0 & 0 & 1 & */p_3 \\ 0 & 0 & 0 & 0 \end{bmatrix}

  2. Use replacements to kill entries above pivots.

[1∗∗01∗000]    →    [10∗01∗000]\begin{bmatrix} 1 & * & * \\ 0 & 1 & * \\ 0 & 0 & 0 \end{bmatrix} \;\;\to\;\; \begin{bmatrix} 1 & 0 & * \\ 0 & 1 & * \\ 0 & 0 & 0 \end{bmatrix}

Theorem

Every matrix AA reduces to a unique matrix in RREF.

Echelon and Solutions

  • If last column of augmented matrix is pivot → no solution.
  • If every column but last is pivot → unique solution.
  • If last col is not pivot and another col is free → infinitely many solutions.

Example

[100∣1010∣−3001∣2]    ⇒    x1=1,  x2=−3,  x3=2\begin{bmatrix} 1&0&0&|&1\\ 0&1&0&|&-3\\ 0&0&1&|&2 \end{bmatrix} \;\;\Rightarrow\;\; x_1=1,\;x_2=-3,\;x_3=2