This calculator converts a linear system into augmented matrix form, performs row reduction, classifies the system, displays solution behavior, and visualizes the output.
Enter Matrix Data
Enter one row per line. Separate values with spaces or commas. Include the constant term as the last column.
Example Data Table
| Equation | Linear Form | Augmented Row |
|---|---|---|
| 1 | 2x + y - z = 8 | [2, 1, -1 | 8] |
| 2 | -3x - y + 2z = -11 | [-3, -1, 2 | -11] |
| 3 | -2x + y + 2z = -3 | [-2, 1, 2 | -3] |
Formula Used
Augmented matrix form: [A | b], where A stores coefficients and b stores constants.
Row reduction: Use elementary row operations until the matrix reaches row echelon form or reduced row echelon form.
Allowed operations:
- Swap two rows: Ri ↔ Rj
- Scale a row: Ri → kRi, where k ≠ 0
- Replace a row: Ri → Ri + kRj
Consistency test: If a row becomes [0 0 … 0 | c] with c ≠ 0, the system is inconsistent.
Rank rule: A unique solution occurs when rank(A) = rank([A|b]) = number of variables.
How to Use This Calculator
- Select the number of equations and variables.
- Enter the augmented matrix with one row per line.
- Place the constant term at the end of each row.
- Click Calculate Matrix Form.
- Review the original matrix, reduced matrix, rank values, and system type.
- Use the CSV and PDF buttons to export the result.
- Inspect the Plotly graph for a visual summary of solution values.
FAQs
1. What is an augmented matrix?
An augmented matrix combines the coefficient matrix and constant column into one rectangular array. It gives a compact way to solve linear systems using row operations.
2. What does the vertical divider mean?
The divider separates variable coefficients from the constants column. Everything before the divider belongs to matrix A, while the final column belongs to vector b.
3. What is RREF?
Reduced row echelon form is a simplified matrix form where each pivot equals one and is the only nonzero entry in its column. It makes solution reading easier.
4. How do I know if the system has no solution?
If row reduction creates a contradictory row such as [0 0 0 | 5], the system is inconsistent. That means no values can satisfy all equations together.
5. What causes infinitely many solutions?
Infinitely many solutions appear when the system is consistent but lacks enough independent pivots. At least one variable becomes free and can take many values.
6. Why does rank matter?
Rank shows how many independent rows or equations are present. Comparing coefficient rank and augmented rank helps classify systems as unique, infinite, or inconsistent.
7. Can this calculator handle non square systems?
Yes. The tool supports rectangular augmented matrices, so you can analyze systems with more equations than variables or more variables than equations.
8. What does the graph represent?
The graph shows solution values or leading numeric terms from the final expressions. It helps compare variable magnitudes quickly after matrix reduction finishes.