Calculator Input
Example Data Table
This sample solves a three variable system using A × X = B.
| Equation | a1 | a2 | a3 | b | Expected value |
|---|---|---|---|---|---|
| 2x + y − z = 8 | 2 | 1 | -1 | 8 | x = 2 |
| -3x − y + 2z = -11 | -3 | -1 | 2 | -11 | y = 3 |
| -2x + y + 2z = -3 | -2 | 1 | 2 | -3 | z = -1 |
Formula Used
Main equation: A × X = B
Unique solution: X = A-1 × B, when det(A) ≠ 0.
Rank test: compare rank(A) with rank([A|B]).
Residual: R = A × X − B. Smaller residuals mean stronger numeric agreement.
The calculator uses pivoted elimination for the actual numeric solve. Partial pivoting reduces round-off errors by choosing the strongest pivot available in each column. The inverse is shown only when the coefficient matrix is nonsingular.
How to Use This Calculator
- Select the square size of matrix A.
- Select how many right side columns matrix B needs.
- Enter each coefficient from your equations into A.
- Enter constants or right side vectors into B.
- Set decimal precision and pivot tolerance if needed.
- Press the solve button and review the result above the form.
- Use CSV or PDF download for reports and records.
Solving Matrix Equations in Study and Work
Why Matrix Form Helps
Matrix equations give a compact way to handle many linked equations. Instead of solving each line by hand, you place coefficients inside matrix A. The unknown values go inside matrix X. The known constants go inside matrix B. This format is clear. It also scales well when the system grows.
What the Calculator Checks
A useful calculator should do more than return numbers. It should also explain whether the answer is valid. This tool checks determinant, rank, inverse availability, residual error, and a condition estimate. These values help you spot singular systems. They also show when a result may be sensitive to small input changes.
When a Unique Solution Exists
A square matrix equation has one unique solution when matrix A has full rank. In that case, the determinant is not zero. The inverse exists. The calculator can then compute X with pivoted elimination. It also verifies the answer by multiplying A and X again.
When Results Need Care
Some systems do not have one clean answer. If the augmented rank is greater than the coefficient rank, the system is inconsistent. No solution can satisfy every row. If both ranks match but stay below the number of variables, the system is dependent. Infinite solutions may exist. In both cases, this tool explains the status instead of forcing a false answer.
Practical Uses
Students can use this page to check algebra homework. Teachers can prepare examples for class. Engineers can inspect small numeric models. Analysts can solve grouped balance equations. The export buttons make it easy to save the entered matrices and computed checks. The residual table is especially useful during review. A near zero residual confirms that the solution fits the original equation within the selected precision.
Frequently Asked Questions
1. What does this calculator solve?
It solves square matrix equations in the form A × X = B. Matrix A stores coefficients, X stores unknowns, and B stores right side values.
2. Can it solve more than one right side?
Yes. You can select up to three columns for B. Each column is treated as a separate right side vector in the same system.
3. What happens when determinant is zero?
A zero determinant means matrix A is singular. The calculator then checks ranks to decide whether the system has no solution or infinitely many solutions.
4. Why is rank important?
Rank shows how many independent equations are present. Comparing rank(A) and rank([A|B]) helps identify unique, dependent, or inconsistent systems.
5. What is the residual matrix?
The residual is A × X − B. Values near zero show that the calculated solution satisfies the original matrix equation closely.
6. What does pivot tolerance mean?
Pivot tolerance decides when a tiny pivot should be treated as zero. Smaller values are stricter, while larger values mark weak pivots earlier.
7. Why use partial pivoting?
Partial pivoting chooses the largest available pivot in a column. This improves numerical stability and reduces avoidable round-off problems.
8. Can I download the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a clean report containing status, matrices, and checks.