Analyze linear systems with dynamic augmented matrix inputs. See elimination steps, RREF output, and ranks. Download clean reports for checking homework, teaching, or audits.
Set dimensions, enter the augmented matrix values, then solve.
This sample represents a 3×3 system with a unique solution. Use the Load Example button to populate a matching matrix instantly.
| Equation | x1 Coefficient | x2 Coefficient | x3 Coefficient | Constant (b) |
|---|---|---|---|---|
| Eq 1 | 2 | 1 | -1 | 8 |
| Eq 2 | -3 | -1 | 2 | -11 |
| Eq 3 | -2 | 1 | 2 | -3 |
The calculator solves linear systems by converting the augmented matrix [A|b] into reduced row echelon form (RREF)
using Gauss-Jordan elimination with partial pivoting.
Rᵢ ↔ Rⱼ (used to place a stronger pivot).Rᵢ = Rᵢ / pivot (makes the pivot equal to 1).Rⱼ = Rⱼ - a·Rᵢ (clears entries above and below pivots).rank(A) and rank([A|b]).rank(A) = rank([A|b]) = number of variables.rank(A) < rank([A|b]).
If the coefficient matrix is square, the calculator also estimates det(A) using elimination and pivot tracking.
This calculator works with an augmented matrix [A|b], where A stores coefficients and b stores constants. Users can set two to six equations and variables, which suits classroom exercises and small analytical models. Accurate entry matters because a sign error can change pivots, ranks, and classification. The input grid supports decimals, rounding, and tolerance control, helping users interpret small coefficients from measured or imported datasets.
The solver applies Gauss Jordan elimination with partial pivoting to produce reduced row echelon form. Partial pivoting chooses the pivot in each column, improving stability and reducing distortion from small divisors. After selecting a pivot, the row is scaled, then elimination clears values above and below it. When step tracking is enabled, each swap, scaling action, and elimination operation is recorded for daily review.
Solution type is determined by comparing rank(A) and rank([A|b]) after reduction. If both ranks equal the variable count, the system has a unique solution. If the ranks match but stay below the variable count, the system has infinite solutions with free variables. If rank(A) is less than rank([A|b]), the system is inconsistent and has no solution. This rank method is dependable when equations often look similar.
The output panel includes metrics that support error checking and professional review. For square coefficient matrices, the calculator estimates det(A), which indicates singularity when it evaluates to zero. In unique solution cases, residual checks compare each computed left side against the entered right side to reveal typing mistakes or rounding. The displayed RREF matrix helps users inspect pivot columns, zero rows, and constants for quick validation without repeating elimination steps.
Reporting features make the calculator practical for homework, tutoring, and operational documentation. CSV export saves the matrix, RREF values, solution classification, and residual data in a structured format for spreadsheet analysis. PDF export captures the result panel for printing, submission, or audit notes. This workflow supports linear cost models, balancing problems, and resource allocation planning while preserving a clear record of inputs, outputs, and decisions.
It combines the coefficient matrix and constants into one grid. The last column stores the right side values, so row operations update every equation consistently.
Partial pivoting chooses a stronger pivot before division. This improves numerical stability, limits rounding error, and reduces failures caused by tiny pivot values.
It compares rank(A) and rank([A|b]). If the augmented rank becomes larger, the reduced system contains a contradiction, so no valid solution exists.
Free variables are columns without pivots in the RREF matrix. They can vary, and pivot variables are written in terms of those free variables.
The determinant is shown when the coefficient matrix is square. It helps identify singular matrices and supports quick interpretation of solution behavior.
Yes. Enable step tracking to see swaps, scaling, and eliminations. You can also export CSV or PDF reports for review, submission, or documentation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.