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Row-echelon Form of a Matrix

Row-Echelon Form of a Matrix
Echelon Form of a Matrix

A matrix form used when solving linear systems of equations.

 

Two properties of matrices in echelon form: rows start with zeros then a leading 1; leading 1s shift right in successive rows.
3×4 matrix in row-echelon form: row1[1,4,-2,0,6], row2[0,1,7,-5,1], row3[0,0,0,1,5]

 

 

See also

Gaussian elimination, reduced row-echelon form

Key Formula

[a11a12a13⋯a1n0a22a23⋯a2n00a33⋯a3n⋮⋮⋮⋱⋮000⋯amn]\begin{bmatrix} \boxed{a_{11}} & a_{12} & a_{13} & \cdots & a_{1n} \\ 0 & \boxed{a_{22}} & a_{23} & \cdots & a_{2n} \\ 0 & 0 & \boxed{a_{33}} & \cdots & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & a_{mn} \end{bmatrix}
Where:
  • aii\boxed{a_{ii}} = Leading entry (pivot) of each row — the first nonzero entry in that row
  • 00 = All entries below each pivot must be zero
  • m×nm \times n = The matrix has m rows and n columns

Worked Example

Problem: Use elementary row operations to convert the following matrix into row-echelon form: [12−1260−1−13]\begin{bmatrix} 1 & 2 & -1 \\ 2 & 6 & 0 \\ -1 & -1 & 3 \end{bmatrix}
Step 1: Eliminate the entry below the first pivot. Subtract 2 times Row 1 from Row 2: R2←R2−2R1R_2 \leftarrow R_2 - 2R_1.
[12−1022−1−13]\begin{bmatrix} 1 & 2 & -1 \\ 0 & 2 & 2 \\ -1 & -1 & 3 \end{bmatrix}
Step 2: Eliminate the other entry below the first pivot. Add Row 1 to Row 3: R3←R3+R1R_3 \leftarrow R_3 + R_1.
[12−1022012]\begin{bmatrix} 1 & 2 & -1 \\ 0 & 2 & 2 \\ 0 & 1 & 2 \end{bmatrix}
Step 3: Eliminate the entry below the second pivot. Subtract 12\frac{1}{2} times Row 2 from Row 3: R3←R3−12R2R_3 \leftarrow R_3 - \tfrac{1}{2}R_2.
[12−1022001]\begin{bmatrix} 1 & 2 & -1 \\ 0 & 2 & 2 \\ 0 & 0 & 1 \end{bmatrix}
Answer: The row-echelon form is [12−1022001]\begin{bmatrix} 1 & 2 & -1 \\ 0 & 2 & 2 \\ 0 & 0 & 1 \end{bmatrix}. Each pivot is to the right of the one above it, and all entries below each pivot are zero.

Another Example

This example differs because the first entry is zero, requiring a row swap before elimination. It also uses an augmented matrix, showing how row-echelon form applies directly to solving a system of equations.

Problem: Convert the following augmented matrix (from a system of 3 equations in 3 unknowns) into row-echelon form: [013524−2649113]\left[\begin{array}{ccc|c} 0 & 1 & 3 & 5 \\ 2 & 4 & -2 & 6 \\ 4 & 9 & 1 & 13 \end{array}\right]
Step 1: The first pivot position has a zero, so swap Row 1 and Row 2 to get a nonzero entry in the top-left: R1↔R2R_1 \leftrightarrow R_2.
[24−26013549113]\left[\begin{array}{ccc|c} 2 & 4 & -2 & 6 \\ 0 & 1 & 3 & 5 \\ 4 & 9 & 1 & 13 \end{array}\right]
Step 2: Eliminate the entry below the first pivot. Subtract 2 times Row 1 from Row 3: R3←R3−2R1R_3 \leftarrow R_3 - 2R_1.
[24−2601350151]\left[\begin{array}{ccc|c} 2 & 4 & -2 & 6 \\ 0 & 1 & 3 & 5 \\ 0 & 1 & 5 & 1 \end{array}\right]
Step 3: Eliminate the entry below the second pivot. Subtract Row 2 from Row 3: R3←R3−R2R_3 \leftarrow R_3 - R_2.
[24−260135002−4]\left[\begin{array}{ccc|c} 2 & 4 & -2 & 6 \\ 0 & 1 & 3 & 5 \\ 0 & 0 & 2 & -4 \end{array}\right]
Answer: The row-echelon form of the augmented matrix is [24−260135002−4]\left[\begin{array}{ccc|c} 2 & 4 & -2 & 6 \\ 0 & 1 & 3 & 5 \\ 0 & 0 & 2 & -4 \end{array}\right]. You can now solve by back-substitution.

Frequently Asked Questions

What is the difference between row-echelon form and reduced row-echelon form?
Row-echelon form (REF) requires that all entries below each pivot are zero, but entries above the pivots can be nonzero. Reduced row-echelon form (RREF) goes further: every pivot must equal 1, and all entries both above and below each pivot must be zero. RREF gives the solution directly, while REF requires back-substitution to finish solving.
Is row-echelon form unique for a given matrix?
No. A matrix can have many different row-echelon forms depending on which row operations you perform. However, the reduced row-echelon form of a matrix is always unique. The pivot positions (which columns contain pivots) are the same regardless of which REF you produce.
What are the three conditions for a matrix to be in row-echelon form?
A matrix is in row-echelon form when: (1) all rows consisting entirely of zeros are at the bottom, (2) the leading entry (pivot) of each nonzero row is strictly to the right of the leading entry in the row above it, and (3) all entries in a column below a pivot are zero.

Row-Echelon Form (REF) vs. Reduced Row-Echelon Form (RREF)

Row-Echelon Form (REF)Reduced Row-Echelon Form (RREF)
PivotsPivots can be any nonzero valueAll pivots must equal 1
Entries above pivotsCan be any valueMust all be zero
Entries below pivotsMust all be zeroMust all be zero
UniquenessNot unique — multiple valid forms existUnique for any given matrix
How to read the solutionRequires back-substitutionSolution can be read directly
MethodGaussian eliminationGauss-Jordan elimination

Why It Matters

Row-echelon form is the foundation of Gaussian elimination, the standard algorithm for solving systems of linear equations in algebra and linear algebra courses. You will use it extensively when finding solutions to systems, determining whether a system has zero, one, or infinitely many solutions, and computing the rank of a matrix. It also appears in applications from engineering to computer science whenever large systems of equations must be solved efficiently.

Common Mistakes

Mistake: Forgetting to check whether the pivot position is zero before proceeding with elimination.
Correction: If a pivot position contains zero, you must swap that row with a lower row that has a nonzero entry in that column. Skipping this step leads to division by zero or an incorrect result.
Mistake: Assuming row-echelon form requires pivots to equal 1.
Correction: Pivots equal to 1 is a requirement of reduced row-echelon form, not row-echelon form. In REF, the pivot just needs to be the first nonzero entry in its row — it can be any nonzero number.

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