Find one special solution for matrix systems. View elimination steps, pivots, rank, and consistency checks. Download reports and reuse sample data for guided practice.
Enter one matrix row per line. Use spaces or commas between values. Enter the right side vector in the same row order.
The calculator solves the linear system A x = b.
It builds the augmented matrix [A|b].
Then it applies elementary row operations until reduced row echelon form appears.
If the system is consistent and has free variables, those free variables are set to 0.
The remaining pivot values create one special solution.
Consistency test: rank(A) = rank([A|b]).
Unique solution test: rank(A) = number of variables.
Infinite solution test: rank(A) < number of variables, while consistency still holds.
| Row | Matrix A Entry Row | Vector b |
|---|---|---|
| 1 | 1, 2, 1 | 4 |
| 2 | 2, 4, 2 | 8 |
| 3 | 1, 1, 0 | 2 |
For this example, one special solution is x1 = 0, x2 = 2, x3 = 0.
A matrix system is written as A x = b. Some systems have one answer. Some have many answers. Some have none. A special solution is one valid answer chosen from a consistent system. It is usually built by setting every free variable to zero. That choice gives a clean starting point. It also makes the final answer easier to read.
This idea appears in algebra, data fitting, control models, and numerical work. It helps you separate pivot variables from free variables. It also helps you detect dependent columns quickly. When a system has infinitely many answers, you often do not want the full parameter form first. A special solution gives one concrete vector immediately. That is helpful for checking homework, testing software, and verifying matrix models.
This calculator reads the coefficient matrix and the right side vector. Then it forms the augmented matrix [A|b]. Next it applies row operations until reduced row echelon form is reached. The pivot columns identify the dependent variables. The free columns show which variables can move. To create one special solution, the free variables are fixed at zero. The pivot values then define the special vector.
Start with the consistency test. If rank(A) and rank([A|b]) are different, the system is inconsistent. No special solution exists. If the ranks match and equal the number of variables, the system has one unique answer. If the ranks match but stay smaller than the number of variables, the system has infinitely many answers. In that case, the shown vector is one special solution only. It is not the full general solution.
The determinant is shown for square matrices. A zero determinant often signals dependence, though rank remains the stronger test. The residual check compares A x with b. Residual values near zero confirm that the displayed vector satisfies the system. You can also review the elimination steps and the final reduced matrix. Those details make the calculation easier to trust and easier to explain in class notes.
Enter one row at a time. Keep the row order of b consistent with A. Use spaces or commas clearly. Double check the declared dimensions before solving. Small input mistakes can change the rank and the final conclusion. The example data in this page gives a safe pattern. Use it first. Then replace it with your own matrix when you are ready.
A special solution is one valid solution chosen from a consistent system with free variables set to zero. It is a clean representative answer for systems with infinitely many solutions.
If rank(A) and rank([A|b]) are different, the system is inconsistent. The calculator reports that no special solution exists for the entered data.
Yes. The matrix does not need to be square. The tool checks dimensions, performs row reduction, and reports consistency, rank, and one special solution when possible.
Free variables appear when a column has no pivot in reduced row echelon form. They can take arbitrary values in the full general solution. This calculator sets them to zero.
The determinant exists only for square matrices. If your matrix is rectangular, the calculator leaves the determinant field unavailable and focuses on rank-based analysis instead.
Yes. You can enter integers, decimals, and negative values. Use spaces or commas between entries and keep one matrix row on each line.
The residual check compares A x with b. If the system is solved correctly, each residual value should be zero or very close to zero after rounding.
No. The special solution is only one member of the full solution set. The general solution also includes the effect of free variables and their parameters.
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