Compute RREF with exact fraction arithmetic
Enter any matrix up to 8×8. The calculator performs Gaussian elimination using exact rational arithmetic — no floating-point rounding. Results show pivot positions, matrix rank, and both REF and RREF forms.
Linear Algebra Guides
24 guidesCommon Mistakes in Gauss-Jordan Elimination (and How to Avoid Them)
Avoid the most common Gauss-Jordan elimination errors: sign mistakes, skipping upward clearing, fraction slip-ups, and zero pivot mishandling.
2026-06-15GuidesUsing RREF to solve systems of linear equations
How to set up an augmented matrix, apply row reduction, and read unique, infinite, and no-solution cases directly from the reduced row echelon form.
2026-06-15GuidesEchelon Form and Back-Substitution vs. Full RREF
Compare Gaussian elimination with back-substitution to Gauss-Jordan reduction to RREF. See both methods on the same system with worked examples.
2026-06-14GuidesHow to Find the Span of a Set of Vectors
Learn how to find the span of a set of vectors using row reduction, check if a vector belongs to a span, and determine if vectors span all of R^n.
2026-06-13GuidesAugmented Matrices: Setting Up a System for Row Reduction
Learn how to build an augmented matrix from a linear system, handle zero coefficients, and interpret the result after row reduction.
2026-06-12GuidesUsing RREF to Solve a 3x3 System of Equations
Step-by-step guide to solving a 3x3 linear system using RREF: build the augmented matrix, row-reduce, and read off x, y, z.
2026-06-11GuidesConsistent vs. Inconsistent Systems: How RREF Tells You
Learn how reduced row echelon form instantly reveals whether a linear system has solutions, and why the no-solution row is the key signal.
2026-06-10GuidesThe Rank-Nullity Theorem with Worked Examples
The rank-nullity theorem states rank(A) + nullity(A) = n. Learn what this means, why it holds, and see two fully worked examples.
2026-06-09GuidesHow to Compute a Determinant via Row Reduction
Learn how row operations affect the determinant, reduce a matrix to triangular form, and compute det step by step with a worked 3x3 example.
2026-06-08GuidesRREF vs REF: what's the difference and when does it matter?
Row echelon form and reduced row echelon form look similar but have a critical difference. Here's what that difference is and which form you actually need.
2026-06-08GuidesRow Space, Column Space, and Null Space: The Big Picture
Learn how row space, column space, and null space relate to every matrix. See how RREF reveals bases for all three fundamental subspaces with a worked example.
2026-06-07GuidesParametric Vector Form of a Solution Set
Learn how to express a solution set in parametric vector form x = p + s·v1 + t·v2, with a fully worked example covering free variables and geometry.
2026-06-06GuidesRREF and Linear Independence: Testing a Set of Vectors
Learn how to test linear independence using RREF. A pivot in every column means independent; a free column reveals the dependency relation.
2026-06-05GuidesSolving Homogeneous Systems (Ax = 0)
Learn how to solve homogeneous systems Ax=0, find nontrivial solutions using free variables, and understand the null space with a worked example.
2026-06-04GuidesHow to Find a Basis for the Column Space
Learn how to find a basis for the column space of a matrix using row reduction and pivot columns, with a fully worked example.
2026-06-03GuidesHow to Find a Basis for the Null Space
Learn how to find a basis for the null space of a matrix step by step: row reduce, identify free variables, and write parametric vector form.
2026-06-02GuidesHow to find RREF by hand: Gauss-Jordan elimination
Step-by-step Gauss-Jordan elimination on a 3×4 augmented matrix, with every row operation shown explicitly so you can follow the same process yourself.
2026-06-01GuidesThe Three Elementary Row Operations
The three elementary row operations (swap, scale, replace), with notation, worked examples, and why each preserves the solution set.
2026-06-01GuidesNo Solution, One Solution, or Infinitely Many: Reading the RREF
Learn how to determine if a linear system has no solution, one unique solution, or infinitely many by reading its reduced row echelon form.
2026-05-30GuidesFinding the Inverse of a Matrix with Gauss-Jordan Elimination
Find a matrix inverse with Gauss-Jordan elimination: a worked 3x3 example, augmented identity setup, and how to spot a singular matrix.
2026-05-28GuidesHow to Find the Rank of a Matrix Using RREF
Learn how to find the rank of a matrix by counting pivot positions in its RREF. Includes two fully worked examples and a clear explanation of rank-deficiency.
2026-05-26GuidesFree Variables and Basic Variables in a Linear System
Identify free and basic variables in a linear system, write parametric solutions, and see why free variables yield infinitely many solutions.
2026-05-24GuidesPivot Positions and Pivot Columns Explained
Learn what pivot positions and pivot columns are in linear algebra, how to identify them in RREF, and what they tell you about a matrix's solutions.
2026-05-22GuidesWhat is Reduced Row Echelon Form (RREF)?
A precise definition of RREF with the four conditions every matrix must satisfy, plus a clear contrast with matrices that fall short of those conditions.
2026-05-22