Pivots of a Matrix in Row Echelon Form

Understanding Row Echelon Form and finding the pivots of a matrix are essential steps in solving systems of linear equations. Review the definitions and properties below, then test your understanding with practice questions and step-by-step solutions.

Note: you can verify your own matrices using our Row Echelon Form Calculator.

Definition of a Matrix in Row Echelon Form and Pivots

A matrix is in row echelon form (REF) if it satisfies the following three conditions:

  1. All rows consisting entirely of zeros are located at the bottom of the matrix.
  2. For any nonzero row, its first non-zero entry is a 1 (often called the leading 1).
  3. For any two successive nonzero rows, the leading 1 in the lower row is positioned strictly to the right of the leading 1 in the upper row.

Definition of pivots. When a matrix is in row echelon form, the pivot positions (or simply pivots) are exactly the positions of the leading 1's in the nonzero rows.

Key idea: the three conditions ensure that the nonzero rows "stair-step" down to the right — each pivot sits strictly to the right of the pivot above it.
Note on uniqueness. A matrix has many row echelon forms (obtained by different sequences of row operations), but only one reduced row echelon form (RREF). If the pivots are also the only nonzero entries in their columns, the matrix is in RREF.
Practical recipe for finding pivots. Scan the rows from top to bottom; in each nonzero row, the pivot is the first non-zero entry (which equals 1).

Examples of Matrices in Row Echelon Form

Example 1 — Pivots in a 4×5 Matrix

\[ \begin{bmatrix} \color{red}{1} & -1 & -2 & 0 & 1 \\ 0 & \color{red}{1} & 3 & 0 & -2 \\ 0 & 0 & \color{red}{1} & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \]

The pivots are:

  • the leading 1 in row 1, column 1;
  • the leading 1 in row 2, column 2;
  • the leading 1 in row 3, column 3.

(The pivot positions are marked in red.)

Example 2 — Pivots in a 4×6 Matrix

\[ \begin{bmatrix} \color{red}{1} & -1 & -2 & 0 & 1 & 0 \\ 0 & 0 & \color{red}{1} & 5 & 1 & 0 \\ 0 & 0 & 0 & 0 & \color{red}{1} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \]

The pivots are:

  • the leading 1 in row 1, column 1;
  • the leading 1 in row 2, column 3;
  • the leading 1 in row 3, column 5.

(The pivot positions are marked in red.)

Practice Questions

Examine the following six matrices. Determine which ones are not in row echelon form and explain why. For those that are in row echelon form, state the positions of their pivots.

1. \[ \begin{bmatrix} 1 & 4 & 3 & 0 & 1 \\ 0 & 1 & -2 & -3 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 6 & 0 \end{bmatrix} \]

2. \[ \begin{bmatrix} 1 & -1 & 2 & -1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix} \]

3. \[ \begin{bmatrix} 1 & 0 & -3 \\ 1 & 1 & 2 \end{bmatrix} \]

4. \[ \begin{bmatrix} 1 & 2 & 0 & -1 & -2 \\ 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \]

5. \[ \begin{bmatrix} 1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \]

6. \[ \begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 2 \end{bmatrix} \]

View Solutions to the Questions
  1. Matrix 1 is not in row echelon form. The leading 1 in row 4 is in column 3, the same column as the leading 1 in row 3 — it is not strictly to the right. This violates condition 3.
  2. Matrix 2 is in row echelon form. It has 2 pivots: at row 1, column 1 and at row 2, column 3.
  3. Matrix 3 is not in row echelon form. The leading 1 in row 2 is in column 1, the same column as the leading 1 in row 1 — it is not strictly to the right. This violates condition 3.
  4. Matrix 4 is in row echelon form. It has 3 pivots: at row 1, column 1; at row 2, column 3; and at row 3, column 4.
  5. Matrix 5 is in row echelon form. It has 2 pivots: at row 1, column 1 and at row 2, column 3.
  6. Matrix 6 is not in row echelon form. Row 1 consists entirely of zeros and must be placed at the bottom. This violates condition 1.

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