RREF Calculator

Row reduction is the workhorse of linear algebra. The same three moves (swap two rows, scale a row, add a multiple of one row to another) solve systems of equations, find the rank of a matrix, test vectors for independence, compute inverses and describe null spaces. By hand it means carrying fractions through a dozen steps, where one slip spoils everything after it. The calculator above performs each step exactly and shows it, so you can compare your work line by line.

Three related tools cover the neighbors. The Matrix Calculator multiplies and inverts matrices and finds determinants and eigenvalues. The System of Equations Calculator accepts equations typed in x, y and z. The Vector Calculator handles dot and cross products. In the text below, matrices are written row by row with semicolons between rows and a bar before the right-hand side, so [1, 2 | 3; 4, 5 | 6] is the augmented matrix of x + 2y = 3 and 4x + 5y = 6. The Paste as text box accepts exactly that notation.

What row echelon form and reduced row echelon form mean

A matrix is in row echelon form (REF) when three conditions hold: rows of all zeros sit at the bottom; the first nonzero entry of each row, called its pivot or leading entry, lies to the right of the pivot in the row above; and every entry below a pivot is zero. Reduced row echelon form (RREF) adds two more: every pivot equals 1, and each pivot is the only nonzero entry in its column. The matrix [1, 2, 0 | 5; 0, 0, 1 | 1; 0, 0, 0 | 0] is in RREF. The matrix [1, −1, 2 | 5; 0, 1, 1/2 | 7/2; 0, 0, 1 | 3] is in REF but not RREF, because the entries above its second and third pivots are not zero.

The distinction matters because of uniqueness. A matrix has many row echelon forms (scale a row, or add a lower row to a higher one, and you have another valid REF), but exactly one reduced row echelon form. That is why answer keys can disagree about an REF yet must agree about the RREF. The REF this calculator reports is the one produced by forward elimination with each pivot scaled to 1, the version many textbooks print.

The three elementary row operations

Every step of row reduction is one of three moves. Each can be undone by another move of the same kind, which is why row operations never change the solution set of a system:

The notation matches the step list: R2 → R2 − 3R1 means "replace row 2 with row 2 minus 3 times row 1". A replacement must keep the row being replaced, with a nonzero coefficient; writing R2 → R1 + R3 is not an elementary operation and can throw information away. The last column explains why row reduction also computes determinants, which the section on determinants below puts to use.

Gauss-Jordan elimination in five steps

1. Start in the top-left corner and find the leftmost column with a nonzero entry at or below the current row.
2. If the current row has a zero in that column, swap it with the first row below that does not.
3. Scale the row so the pivot becomes 1.
4. Add multiples of the pivot row to each row below until every entry under the pivot is zero. Move one row down and one column right, and repeat until you run out of rows or columns. The matrix is now in REF.
5. Starting from the last pivot and working upward, add multiples of each pivot row to the rows above it until every entry above the pivot is zero. The matrix is now in RREF.

Gaussian elimination stops after step 4 and finishes with back substitution in the equations. Gauss-Jordan elimination continues to the RREF so the answer can be read directly from the matrix. The calculator follows this hand method exactly and never reorders rows except to replace a zero pivot, so its steps look like what you would write on paper.

Worked example 1: three equations with exactly one solution

Solve 2y + z = 7, x − y + 2z = 5 and 2x + y − z = 1 (the calculator's default). The augmented matrix is [0, 2, 1 | 7; 1, −1, 2 | 5; 2, 1, −1 | 1]. Its top-left entry is 0, so the first move has to be a swap. The table lists all eight row operations exactly as the calculator reports them:

The last column now reads x = 1, y = 2, z = 3. Substituting into the original equations confirms it: 2·2 + 3 = 7, 1 − 2 + 2·3 = 5 and 2·1 + 2 − 3 = 1. Steps 3 and 4 create the fractions 1/2, 7/2, −13/2 and −39/2. Halves survive a decimal calculator, but if a pivot had been 3 or 7, decimals would have rounded at that point and carried the error forward; exact fractions keep every value until it cancels.

Worked example 2: infinitely many solutions and free variables

Now take x + 2y − z = 4, 2x + 4y + z = 11 and 3x + 6y = 15 (the Infinitely many preset). Five operations produce the RREF [1, 2, 0 | 5; 0, 0, 1 | 1; 0, 0, 0 | 0]. Columns 1 and 3 hold pivots and column 2 does not, so y is a free variable and x and z are basic. The bottom row reads 0 = 0 and carries no information, because the third equation is simply the sum of the first two. The two remaining rows say x + 2y = 5 and z = 1. Setting y = t gives

Every real number t produces a solution: t = 0 gives (5, 0, 1) and t = 1 gives (3, 1, 1), and both satisfy all three original equations. Geometrically, the three planes meet in a line, and the vector form describes it as a point on the line plus multiples of its direction (−2, 1, 0).

Worked example 3: a system with no solution

The No solution preset is x + y + 2z = 3, 2x + 3y + z = 4 and 3x + 4y + 3z = 9. After three replacements the third row is [0, 0, 0 | 2], which says 0x + 0y + 0z = 2. No choice of x, y and z can make that true, so the system is inconsistent. The cause is visible in the original equations: adding the first two gives 3x + 4y + 3z = 7, which contradicts the third equation's 9. The calculator still finishes the job, scaling that row and reaching the RREF [1, 0, 5 | 0; 0, 1, −3 | 0; 0, 0, 0 | 1]. The pivot in the last column is the formal signal: rank(A) = 2 but rank([A | b]) = 3.

Rank, pivot columns and the number of solutions

Three numbers decide every case: the rank of the coefficient matrix A, the rank of the augmented matrix [A | b], and the number of unknowns n. If rank(A) < rank([A | b]), there is no solution. If rank(A) = rank([A | b]) = n, there is exactly one. If rank(A) = rank([A | b]) < n, there are infinitely many, with n − rank(A) free variables. Each system below was row-reduced with the calculator.

The fourth and fifth systems differ in a single number and land on opposite sides of the rule: three equations in two unknowns can be consistent, but only when the third equation agrees with the first two. The last system is homogeneous (every right-hand side is 0), so it always has at least the zero solution, and its free variable spans the null space of the coefficient matrix.

Determinants and inverses from the same row operations

Row reduction does more than solve systems. In worked example 1 the only operations that change a determinant are the swap (factor −1) and the two scalings (factors 1/2 and −2/13). The coefficient part ends as the identity matrix, whose determinant is 1, so det(A) × (−1) × (1/2) × (−2/13) = 1, which gives det(A) = 13. To invert a matrix, row-reduce [A | I]. For the same coefficient matrix A = [0, 2, 1; 1, −1, 2; 2, 1, −1], eight operations produce A⁻¹ = [−1/13, 3/13, 5/13; 5/13, −2/13, 1/13; 3/13, 4/13, −2/13], and multiplying A⁻¹ by (7, 5, 1) returns (1, 2, 3) again. You can enter the 3 × 6 matrix [A | I] here with the augmented box unticked, or use the inverse operation of the Matrix Calculator.

How to use this calculator, and who it is for

Set the number of rows and columns (up to 6 × 7). Leave Augmented matrix ticked when the last column holds the right-hand sides of equations; untick it to analyze a plain matrix, and the calculator reports rank, pivot columns, nullity and a basis for the null space instead of solutions. Type entries as integers, decimals or fractions such as −3/4, and leave a cell empty for 0. Paste as text accepts a whole matrix, one row per line, with an optional bar before the last entry; write numbers without thousands separators (1000, not 1,000). The results show the RREF with its pivots underlined, the verdict, the REF, and the numbered row operations with the changed rows highlighted. Switching to decimals changes only the display; the arithmetic stays exact, and a nonzero entry too small for the chosen decimal places appears in e-notation, such as 3.333e-5, rather than as 0.

It is built for students in a first linear algebra course who want to check homework one row operation at a time, for teachers who need worked examples with predictable fractions, and for anyone who has to solve a small linear system quickly, such as balancing a chemical equation, splitting a budget across several constraints or fitting a polynomial through a few points.

Key terms

• Pivot (leading entry): the first nonzero entry of a row in echelon form; in RREF it is always 1.
• Pivot column: a column that contains a pivot; the matching variable is a basic variable.
• Free variable: a variable whose column has no pivot; it can take any value, and the basic variables are written in terms of it.
• Rank: the number of pivots, equal to the number of nonzero rows in echelon form.
• Nullity: the number of free columns, n − rank, which is the dimension of the null space.
• Consistent and inconsistent: a system with at least one solution, and a system with none.
• Homogeneous system: Ax = 0, which always has the zero solution.
• Row equivalent: two matrices that can be turned into each other by row operations; they share the same RREF and the same solution set.

Tips for row reducing by hand

• If some row starts with 1, swap it to the top first; it keeps fractions out of the first column. The calculator does not do this, so its steps may show fractions where yours do not, but the RREF will match.
• Clear one column completely before moving to the next, and always apply the operation to the whole row, including the entry after the bar.
• Divide out common factors early: if every entry of a row is a multiple of 3, scale it by 1/3 before using it.
• A row that turns into all zeros is not an error; it marks a dependent equation. Keep going.
• Check the final answer by substituting into the original equations, not the reduced ones, since a slip in an early step carries into every later matrix.

Limitations

The calculator handles up to six rows and seven columns, enough for six equations in six unknowns, and every entry must be a number. It deliberately follows the hand method rather than the partial pivoting used in numerical software, which picks the largest available entry as each pivot to limit rounding error. With exact fractions there is no rounding error to limit, but it does mean the intermediate steps can differ from software that pivots, even though the final RREF is identical. For an inconsistent system it reports that no solution exists; it does not compute a least-squares compromise.

Related calculators

• For products, inverses, determinants and eigenvalues, use the Matrix Calculator.
• To solve two or three equations typed in x, y and z, use the System of Equations Calculator.
• For dot products, cross products and the angle between vectors, use the Vector Calculator.
• For a single equation in one unknown, the Linear Equation Calculator is faster.
• To simplify the fractions that row reduction produces, use the Fraction Calculator, and to find a common multiple for clearing denominators, the LCM Calculator.

Sources

Row reduction is a deterministic algorithm, so the steps, matrices and tables above were computed directly by applying it to the matrices shown. The definitions follow these references:

• MIT OpenCourseWare 18.06SC Linear Algebra: Elimination with Matrices, pivots, row exchanges and the elimination steps.
• MIT OpenCourseWare 18.06SC: Solving Ax = 0, Pivot Variables, Special Solutions, the reduced row echelon form R, rank as the number of pivots, free columns and special solutions.
• MIT OpenCourseWare 18.06SC: Solving Ax = b, Row Reduced Form R, all solutions of Ax = b from the free variables of the reduced form.
• NIST Digital Library of Mathematical Functions, §1.3 Determinants: a row swap changes the sign of a determinant, scaling a row scales it, and adding a multiple of one row to another leaves it unchanged.
• NIST Digital Library of Mathematical Functions, §1.2 Elementary Algebra: square matrices, the inverse and the condition det(A) ≠ 0 (§1.2(vi)).