Calculator Form
Example Data Table
| System | Equations | Variables | Coefficient Rank | Augmented Rank | Result |
|---|---|---|---|---|---|
| x + y = 5, x - y = 1 | 2 | 2 | 2 | 2 | Unique solution |
| x + y = 4, 2x + 2y = 8 | 2 | 2 | 1 | 1 | Infinite solutions |
| x + y = 4, 2x + 2y = 9 | 2 | 2 | 1 | 2 | No solution |
Formula Used
A linear system can be written as A x = b. Here, A is the coefficient matrix. The vector x contains the unknown variables. The vector b contains the constants.
The augmented matrix is written as [A | b]. The calculator applies reduced row echelon form. It then compares two ranks.
If rank(A) is not equal to rank([A | b]), the system is inconsistent. If rank(A) equals rank([A | b]) and also equals the number of variables, the system has one unique solution. If rank(A) equals rank([A | b]) but is less than the variable count, the system has infinitely many solutions.
Free variables are calculated as: number of variables minus rank(A).
How to Use This Calculator
- Enter the number of equations.
- Enter the number of variables.
- Click Update Matrix Size when dimensions change.
- Enter all coefficients for each equation.
- Enter each constant in the final field.
- Click Calculate Consistency.
- Review ranks, row steps, and solution type.
- Use CSV or PDF download for records.
Linear Algebra Consistency Calculator Guide
Purpose of the Tool
This calculator studies systems of linear equations. It checks whether equations can work together. A system is consistent when at least one solution exists. It is inconsistent when the equations contradict each other. The tool is useful for algebra, matrices, engineering models, economics, statistics, and applied mathematics.
Rank Based Testing
The main test compares matrix ranks. The coefficient matrix contains only variable coefficients. The augmented matrix also contains constants. When both ranks match, the system is consistent. When the augmented rank is larger, a contradiction exists. This contradiction often appears as a zero coefficient row with a nonzero constant.
Unique and Infinite Results
A consistent system can still have different solution types. If the coefficient rank equals the number of variables, every variable has a pivot. The result is one unique solution. If the rank is smaller, some variables do not have pivots. Those variables are free. The system then has infinitely many solutions.
Row Reduction Method
The calculator uses row reduction. It searches for pivots, swaps rows when needed, scales pivot rows, and removes matching entries above and below each pivot. This creates reduced row echelon form. The final matrix makes the rank, pivots, and contradictions easier to read.
Square Matrix Insight
When the coefficient matrix is square, the determinant is also reported. A nonzero determinant means the coefficient matrix has full rank. In that case, a unique solution normally exists for any constant vector. A zero determinant means the matrix is singular, so the system needs rank comparison.
Practical Value
This tool saves time during homework, model checking, and data preparation. It also helps users verify manual elimination steps. The downloadable reports make results easier to share. Students can compare their work with the row operations. Professionals can quickly test whether a model has enough independent equations.
FAQs
1. What does system consistency mean?
Consistency means the system has at least one solution. The equations do not contradict each other. A consistent system may have one solution or infinitely many solutions.
2. What is an inconsistent system?
An inconsistent system has no solution. This happens when the augmented matrix creates a contradiction, such as zero variables equaling a nonzero constant.
3. What is the coefficient matrix?
The coefficient matrix contains only the numbers multiplying the variables. It does not include the constants from the right side of the equations.
4. What is the augmented matrix?
The augmented matrix joins the coefficient matrix with the constants column. It represents the full system in one matrix table.
5. Why are ranks compared?
Ranks show the number of independent rows or equations. Comparing coefficient rank and augmented rank reveals whether the constants create a contradiction.
6. When is the solution unique?
The solution is unique when the system is consistent and the coefficient rank equals the number of variables. Then every variable has a pivot.
7. What are free variables?
Free variables are variables without pivot columns. They can take many values, which creates infinitely many solutions in a consistent dependent system.
8. Can this calculator handle rectangular systems?
Yes. It supports systems with different numbers of equations and variables. Rank comparison works for square and rectangular systems.