Enter Your 3x4 Augmented Matrix
Columns one through three are x, y, and z coefficients. The last column contains constants. Fractions such as 1/2 are accepted.
Example Data Table
| Example | Augmented matrix | Expected result |
|---|---|---|
| Unique solution | [2, 1, -1 | 8] | x = 2, y = 3, z = −1 |
| Infinite solutions | [1, 1, 1 | 3] | One free parameter |
| No solution | [1, 1, 1 | 2] | Inconsistent rows |
Formula Used
A 3x4 augmented matrix stores a system of three equations:
[a21 a22 a23 | b2]
[a31 a32 a33 | b3]
The calculator uses elementary row operations:
- Swap two rows:
Ri ↔ Rj. - Multiply a row by nonzero k:
Ri → kRi. - Add a multiple of one row to another:
Ri → Ri + kRj.
Classification uses ranks. If rank(A) differs from rank([A|b]), no solution exists. Equal rank three gives one solution. Equal lower ranks give infinitely many solutions.
How to Use This Calculator
- Write each equation using x, y, and z in the same order.
- Enter its three coefficients and final constant across one row.
- Choose REF, RREF, or both, then choose rounding precision.
- Select Solve Matrix. The result appears above the entry grid.
- Review ranks, transformed matrices, solution lines, and row operations.
- Use CSV or PDF downloads to save your calculation record.
Matrix Guide
Understand the Matrix
A 3x4 augmented matrix represents three linear equations with three unknowns. The first three columns hold coefficients. The last column holds constants. Each row describes one equation. For example, [2, 1, -1 | 8] means 2x + y - z = 8. Matrix form keeps the system compact. It makes row operations easier to track.
Why Row Reduction Helps
Gaussian elimination transforms the matrix into row echelon form. It creates zeros below each pivot. Back substitution then finds unknown values. Gauss-Jordan elimination continues until reduced row echelon form appears. It creates zeros above and below every pivot. The matrix often reveals the answer immediately. A pivot in every variable column indicates a unique solution. A zero coefficient row with a nonzero constant indicates inconsistency. Missing pivots can indicate infinitely many solutions.
What the Calculator Shows
This calculator accepts integers, decimals, and fractions. It calculates the determinant of the coefficient matrix. It also finds the coefficient rank and augmented rank. Those ranks classify the system. Equal full ranks produce one solution. Equal partial ranks produce a solution family. Different ranks produce no solution. The result panel shows the transformed matrix, row-operation record, and solution statement. Choose a precision level for readable values. Select REF, RREF, or both outputs. The calculator keeps tiny rounding errors near zero.
Checking a System
Always enter coefficients in the correct order. Keep x, y, and z columns consistent. Put the equation constant in the final column. Review negative signs before solving. A wrong sign can change the classification. Use the displayed solution to substitute back into original equations. Each equation should balance. For infinite solutions, choose parameter values and test them. For no solution, compare the conflicting equations. They usually describe parallel planes without a shared point.
Practical Uses
Three-variable systems appear in engineering, finance, graphics, and science. They can represent force balance, mixture quantities, circuit currents, or production limits. A matrix calculator reduces repetitive arithmetic. It does not replace interpretation. Units still matter. Assumptions still matter. Rounding may slightly affect close results. Increase precision when values are very small or nearly dependent. Save the CSV or PDF record when documenting work. The example table provides values for practice. Use random entries to explore outcomes. The row steps make each result easier to verify manually.
Frequently Asked Questions
1. What is a 3x4 augmented matrix?
It has three equation rows and four columns. The first three columns contain variable coefficients. The final column contains constants from the right side of each equation.
2. What do the vertical bar and final column mean?
The bar separates coefficient values from constants. It is a compact way to write a linear system without repeatedly writing x, y, z, and equal signs.
3. Can I enter fractions?
Yes. Enter simple fractions such as 1/2, -3/4, or 7/5. The calculator converts them to numerical values before performing row operations.
4. What is the difference between REF and RREF?
REF creates zeros below pivots. RREF also creates zeros above pivots and makes each pivot equal one. RREF normally makes solutions easier to read.
5. Why does the calculator show ranks?
Ranks test consistency. Different coefficient and augmented ranks mean no solution. Equal ranks identify whether the system has a unique solution or infinitely many solutions.
6. Does a zero determinant always mean no solution?
No. A zero determinant means the coefficient matrix is singular. The system may have infinitely many solutions or no solution. Rank comparison determines which case occurs.
7. Why are tiny values displayed as zero?
Computer arithmetic can create extremely small rounding residues. The selected tolerance treats values near zero as zero. This keeps row-reduced output understandable and stable.
8. Can this calculator solve non-square systems?
This page is designed for exactly three equations and three unknowns, stored as a 3x4 augmented matrix. Other dimensions require a different calculator setup.
9. What does a free parameter mean?
A free parameter appears when a variable column lacks a pivot. That variable can take arbitrary values, creating an infinite family of solutions.
10. How can I check a unique solution?
Substitute x, y, and z back into all original equations. Each left side should equal its matching constant. This independently verifies the computed answer.
11. What do CSV and PDF downloads include?
They include the input matrix, rank values, determinant, classification, solution information, selected reduced matrix, and recorded row operations for convenient documentation.