Quick answer
Gauss–Jordan elimination reduces a matrix to reduced row echelon form, and the number of non-zero rows is the rank. An invertible 3 × 3 matrix reduces to the identity, rank 3.
Enter a matrix with rows separated by semicolons. The calculator returns the reduced row echelon form and the rank.
Related tools: Vertex Form Calculator: Convert y = ax² + bx + c to vertex form. Standard Form Calculator: Convert scientific notation to a standard decimal number. Slope-Intercept Form Calculator: Line equation y = mx + b from a slope and a point.
How to use the Row Echelon Form Calculator
- Type matrix rows separated by semicolons, for example 1 2 3; 4 5 6; 7 8 10.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Matrix rows separated by semicolons | Text, such as 1 2 3; 4 5 6; 7 8 10 | — |
Formula and method
Formula
Gauss–Jordan elimination
IEEE-754 double precision; shown to the decimals indicated.
Computed in floating point with a small tolerance; results are shown to six decimals.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Matrix rows separated by semicolons = 1 2 3; 4 5 6; 7 8 10.
| Result | Value |
|---|---|
| Reduced row echelon form | 1 0 0; 0 1 0; 0 0 1 |
| Rank | 3 |
Example 2
Inputs: Matrix rows separated by semicolons = 2 4 6; 1 2 3.
| Result | Value |
|---|---|
| Reduced row echelon form | 1 2 3; 0 0 0 |
| Rank | 1 |
Example 3
Inputs: Matrix rows separated by semicolons = 1 1 1 6; 0 2 5 -4; 2 5 -1 27.
| Result | Value |
|---|---|
| Reduced row echelon form | 1 0 0 5; 0 1 0 3; 0 0 1 -2 |
| Rank | 3 |
The first line is the headline result (reduced row echelon form). Below it the row Echelon Form Calculator lists rank, which describe the same case from other angles. IEEE-754 double precision; shown to the decimals indicated.
Row operations
Swap rows, scale a row and add a multiple of one row to another. Each pivot becomes 1 with zeros above and below. Tiny pivots are treated as zero to avoid rounding noise.
Uses
Solve linear systems (reduce the augmented matrix), find the rank, test whether vectors are independent and compute inverses. A rank below the number of rows means the matrix is singular.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Matrix rows separated by semicolons: left empty shows "Enter matrix rows separated by semicolons." beside that field and no result.
Choosing between the Row Echelon Form Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Vertex Form Calculator](/math/vertex-form/) | a, b and c | Vertex (h, k) |
| [Standard Form Calculator](/math/standard-form-calculator/) | coefficient and exponent | Standard form |
| [Slope-Intercept Form Calculator](/math/slope-intercept-form/) | slope, point x and point y | Equation |
| [Point-Slope Form Calculator](/math/point-slope-form/) | slope, point x₁ and point y₁ | Point-slope form |
| [Standard Form to Slope-Intercept Form Calculator](/math/standard-form-to-slope-intercept-form/) | a, b and c | Slope-intercept form |
Background: calculus, vectors and matrices
A derivative measures how fast a function's output changes with its input, and an integral accumulates a quantity such as area. The determinant of a 2×2 matrix [[a, b], [c, d]] is ad − bc, so [[3, 8], [4, 6]] has determinant 3×6 − 8×4 = −14.
The dot product of two vectors is the sum of the products of matching components and is a single number; a positive value means the vectors point in broadly the same direction.
Assumptions and limitations
- Tiny pivots are treated as zero
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for matrix rows separated by semicolons to confirm this.
Frequently asked questions
What is rank?
The number of independent rows.
What does a zero row mean?
A dependent equation.
Is RREF unique?
Yes, for a given matrix.
What does the Row Echelon Form Calculator calculate?
Reduced row echelon form and rank of a matrix. It returns Reduced row echelon form and Rank.
What formula does the Row Echelon Form Calculator use?
Gauss–Jordan elimination. Computed in floating point with a small tolerance; results are shown to six decimals.
What do I need to enter in the row Echelon Form Calculator?
You enter Matrix rows separated by semicolons.
How is the result rounded or limited?
IEEE-754 double precision; shown to the decimals indicated. Tiny pivots are treated as zero.
What is an example of a row Echelon Form result?
With matrix rows separated by semicolons 1 2 3; 4 5 6; 7 8 10, the reduced row echelon form is 1 0 0; 0 1 0; 0 0 1.
Where does the method come from?
The method follows: OpenStax, Algebra and Trigonometry 2e (Rice University): matrices, determinants and systems of equations (Gaussian elimination).
Related calculators
- Vertex Form Calculator — Convert y = ax² + bx + c to vertex form.
- Standard Form Calculator — Convert scientific notation to a standard decimal number.
- Slope-Intercept Form Calculator — Line equation y = mx + b from a slope and a point.
- Point-Slope Form Calculator — Point-slope and slope-intercept equations of a line.
- Standard Form to Slope-Intercept Form Calculator — Convert Ax + By = C to y = mx + b.
- Fraction Calculator — Add, subtract, multiply and divide fractions with simplified and mixed-number results.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): matrices, determinants and systems of equations (Gaussian elimination)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.