Calculator Input
Example Data Table
| System type | Augmented matrix rows | Rank result | Conclusion |
|---|---|---|---|
| Unique solution | 1 1 32 -1 0 |
rank(A) = rank([A|b]) = 2 | Consistent with one solution. |
| Infinite solutions | 1 2 32 4 6 |
rank(A) = rank([A|b]) = 1 | Consistent with free variables. |
| No solution | 1 2 32 4 8 |
rank(A) = 1, rank([A|b]) = 2 | Inconsistent system. |
Formula Used
For a linear system Ax = b, compare the rank of the coefficient matrix
A with the rank of the augmented matrix [A|b].
rank(A) = rank([A|b])means the system is consistent.rank(A) < rank([A|b])means the system is inconsistent.rank(A) = rank([A|b]) = number of variablesmeans one solution.rank(A) = rank([A|b]) < number of variablesmeans infinitely many solutions.
How to Use This Calculator
- Choose the number of equations.
- Choose the number of variables.
- Enter the augmented matrix row by row.
- Place the constant column at the end.
- Adjust tolerance for decimal or measured values.
- Press calculate to view consistency, ranks, pivots, and RREF.
- Use the CSV or PDF buttons to save the report.
Understanding Matrix Consistency
What Consistency Means
A matrix is consistent when its linear system has at least one solution. The answer depends on the relationship between the coefficient matrix and the augmented matrix. This calculator checks that relationship with row reduction. It also reports ranks, pivot columns, free variables, and a clear decision.
Why The Test Matters
Consistency is useful in algebra, engineering, economics, statistics, and modeling. Many real problems become systems of equations. A consistent system means the constraints can exist together. An inconsistent system means at least one condition conflicts with the others. The conflict usually appears as a row like zero equals a nonzero constant.
Rank Rule
The rank test is the main rule. First, find the rank of the coefficient matrix. Then, find the rank of the augmented matrix. If both ranks match, the system is consistent. If the augmented rank is larger, the system is inconsistent. When the ranks match the number of variables, the solution is unique. When the ranks match but are smaller than the number of variables, infinitely many solutions exist.
RREF Method
Row reduction makes the test practical. The calculator converts the augmented matrix to reduced row echelon form. Pivot columns show leading variables. Non-pivot columns show free variables. These details explain why the final status was chosen. They also help students check each step in a homework solution.
Tolerance And Precision
Tolerance matters because decimal entries can create tiny rounding errors. A value near zero may actually be numerical noise. Set a small tolerance for exact-looking data. Use a larger tolerance when measurements are rounded or experimental. Precision controls how many digits appear in the final report.
Best Practice
Use this tool for quick verification and learning. Enter each augmented row on a separate line. Put coefficients first, then the constant term last. Choose the number of equations and variables. Press calculate to view ranks, RREF, solution type, and exports. The example table shows common outcomes.
Advanced Checks
For advanced checks, compare several inputs. Change one constant and watch the ranks update. This shows how sensitive a system can be. Square systems may also include a determinant clue. A nonzero determinant confirms a unique solution. A zero determinant needs the full rank test before any decision is safe for real matrices.
FAQs
1. What does a consistent matrix mean?
A consistent matrix represents a linear system with at least one solution. It may have one solution or infinitely many solutions, depending on rank and pivot columns.
2. What makes a matrix inconsistent?
A matrix is inconsistent when row reduction creates a contradiction. A common form is a row with all zero coefficients but a nonzero constant.
3. Why compare rank(A) and rank([A|b])?
The comparison shows whether the constants conflict with the coefficient equations. Equal ranks mean consistency. A larger augmented rank means no solution.
4. Can a consistent system have infinite solutions?
Yes. Infinite solutions occur when the ranks match but are smaller than the number of variables. At least one variable becomes free.
5. What is RREF?
RREF means reduced row echelon form. It is a simplified matrix form that reveals pivots, free variables, contradictions, and solution structure.
6. What tolerance should I use?
Use a very small tolerance for exact values. Use a slightly larger tolerance for measured or rounded decimal data to avoid false tiny pivots.
7. Does a nonzero determinant prove consistency?
For a square coefficient matrix, a nonzero determinant gives a unique solution. Therefore, the system is consistent for any constant vector.
8. Can I export the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a clean printable report with ranks and status.