--- title: "Gaussian elimination and row echelon form" description: "Reduce a system with row swaps, nonzero row scaling and row replacement to echelon or reduced echelon form, then read pivots, free variables and the full solution set." canonical: https://lightmysky.com/learn/mathematics/gaussian-elimination-and-row-echelon-form-mt_PZBce3UvRm source: https://lightmysky.com/learn/mathematics/gaussian-elimination-and-row-echelon-form-mt_PZBce3UvRm.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Gaussian elimination and row echelon form Reduce a system with row swaps, nonzero row scaling and row replacement to echelon or reduced echelon form, then read pivots, free variables and the full solution set. Subject: Mathematics · Area: Linear Algebra · Ages 18 to 19 Page: https://lightmysky.com/learn/mathematics/gaussian-elimination-and-row-echelon-form-mt_PZBce3UvRm ## Ready when they can - Carry a system to row echelon form and recognize reduced row echelon form - Identify pivot and free columns and write the general solution - Say why each row operation preserves the solution set ## Lesson: Staircases that solve systems Elimination uses three moves: swap two rows, scale a row by a nonzero number, or add a multiple of one row to another. Each move is reversible, so the solution set never changes. Each matrix row is an equation. Columns before the bar hold coefficients; the bar separates the constants. Work top to bottom toward echelon form, also called staircase form, then back substitute from the last pivot up. **Example.** Solve x + y = 5 and x - y = 1. Add the rows and y cancels: 2x = 6, so x = 3. Then y = 2, and both equations hold. Adding was the allowed move of adding one row to another. In echelon form, zero rows come last. Each pivot is the first nonzero entry of a nonzero row and lies to the right of the pivot above. Reduced form also makes each pivot 1 and the only nonzero entry in its column. In [1 0 2 | 3; 0 1 -1 | 4; 0 0 0 | 0], coefficient columns 1 and 2 have pivots; column 3 is free. The last column holds constants, not a variable. For a consistent system, free variables equal unknowns minus pivots: 3 minus 2 gives 1 here. *(drawing: The row 0 = 5 cannot be true, so the system has no solution.)* **Example.** The augmented matrix of x + y + z = 5, 2x + 2y + 2z = 10 and x - y = 1, with rows R1, R2, R3, is [1 1 1 | 5; 2 2 2 | 10; 1 -1 0 | 1]. R2 <- R2 - 2R1 gives [0 0 0 | 0]. R3 <- R3 - R1 gives [0 -2 -1 | -4]. Swap R2 and R3. Column 3 has no pivot, so z = t. Then -2y - t = -4 gives y = 2 - t/2, and R1 gives x = 3 - t/2. To clear an entry below a pivot, divide that entry by the pivot and subtract that multiple of the pivot row. *(drawing: These moves change the display while keeping the same solution set.)* A row 0 = 5 is a contradiction, so no solution exists. A row 0 = 0 is redundant, but it does not by itself prove infinitely many solutions; the system must be consistent and have a free variable. Scaling by a nonzero number c is undone by scaling by 1/c, so scaling by 4 is undone by 1/4. Scaling a row by zero is not allowed because it loses information and cannot be reversed. **Recap.** Swap, scale, and add toward staircase form, then read pivots, free columns, and the solution. *Original lesson and assessment text by the Open Source Taxonomy project.* ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Systems of linear equations and their solution sets](https://lightmysky.com/learn/mathematics/systems-of-linear-equations-and-their-solution-sets-mt_sQL6JfR7Te) ## Optional connections - [Rearranging formulae when the subject appears twice](https://lightmysky.com/learn/mathematics/rearranging-formulae-when-the-subject-appears-twice-mt_Velmpnxfcj) ## Opens up - [Matrix multiplication and what it represents](https://lightmysky.com/learn/mathematics/matrix-multiplication-and-what-it-represents-mt_0fJOMPlCxr) - [The Inverse of a Matrix and When It Exists](https://lightmysky.com/learn/mathematics/the-inverse-of-a-matrix-and-when-it-exists-mt_1K6lcfMQ_b) - [LU Factorisation, Pivoting and the Cost of a Solve](https://lightmysky.com/learn/mathematics/lu-factorisation-pivoting-and-the-cost-of-a-solve-mt_5Gg2v3Oxrr) - [Linear Independence, Span and Basis](https://lightmysky.com/learn/mathematics/linear-independence-span-and-basis-mt_5RGbip9yC9) - [Conditioning, Stability and Floating-Point Error](https://lightmysky.com/learn/mathematics/conditioning-stability-and-floating-point-error-mt_q8IoCElXIV) - [Linear Regression by Least Squares](https://lightmysky.com/learn/computing/linear-regression-by-least-squares-mt_QRRkpIrwWs)