Enter coefficients, constants, and dimensions for accurate solutions. Review determinant, rank, inverse, and consistency instantly. Download reports and verify every computed matrix result quickly.
| Equation | x1 | x2 | x3 | Constant |
|---|---|---|---|---|
| 1 | 2 | -1 | 3 | 9 |
| 2 | 1 | 1 | 1 | 6 |
| 3 | 3 | 2 | -2 | 1 |
Standard matrix form: A × x = b
Coefficient matrix: A = [aij]
Unknown vector: x = [x1, x2, x3, ... , xn]ᵀ
Constant vector: b = [b1, b2, b3, ... , bn]ᵀ
Determinant rule: If det(A) ≠ 0, the system has a unique solution.
Inverse method: x = A⁻¹b when the inverse exists.
Rank test: If rank(A) = rank([A|b]) = n, the solution is unique.
Consistency rule: If rank(A) = rank([A|b]) < n, infinitely many solutions exist. If ranks differ, no solution exists.
A matrix solver with constant coefficients helps you solve linear systems quickly. It turns many equations into one structured matrix problem. This saves time during homework, test practice, and engineering checks. It also reduces manual algebra mistakes. You can inspect determinant, rank, inverse, and reduced row echelon form in one place.
Constant coefficient systems appear in algebra, statistics, physics, economics, and computer science. They model balances, flows, transformations, and constraints. A clear coefficient matrix shows how every variable contributes to each equation. The constant vector stores fixed outcomes. Together, they define whether a system is consistent, singular, underdetermined, or uniquely solvable.
This calculator does more than basic substitution. It applies row reduction to build the reduced row echelon form. It compares the rank of the coefficient matrix with the augmented matrix. That reveals whether the system has one solution, no solution, or infinitely many solutions. It also verifies answers using the residual vector.
The determinant is a quick signal. A nonzero determinant means the matrix is invertible. Then the system has one exact solution. A zero determinant means the inverse does not exist. In that case, the solver still studies rank and consistency. This is useful when dependent equations or contradictions appear in the system.
Students can compare class methods with computed matrix steps. Teachers can demonstrate elimination clearly. Analysts can export a clean summary for reports. The example table helps you test the page immediately. The step output also supports debugging when data entry errors create unexpected solutions or unstable matrix behavior.
A strong matrix workflow should be accurate, readable, and easy to verify. This tool keeps the layout simple. It shows results above the form for quick review. It also supports CSV and PDF downloads. That makes the calculator useful for study sessions, documentation, and repeated equation solving work.
It means each equation uses fixed numeric coefficients for the variables. The solver reads those fixed values into a coefficient matrix and solves the matching constant vector.
A unique solution exists when the coefficient matrix has full rank. For a square system, that usually means the determinant is not zero and the inverse exists.
A zero determinant means the rows or columns are linearly dependent. The matrix becomes singular, so inverse-based solving is impossible.
Rank is the number of independent rows or pivot positions. It helps identify whether equations are independent, repeated, or inconsistent.
Residuals compare computed results with the original constants. Small residuals confirm the solution satisfies the entered equations within numerical precision.
Yes. The calculator accepts integers, negative numbers, and decimals. You can also adjust the displayed decimal precision for cleaner output.
RREF shows the simplified matrix after elimination. It makes pivot columns, free variables, contradictions, and solved values much easier to interpret.
Infinite solutions occur when the system is consistent but lacks enough independent equations. Some variables become free, so many valid answers satisfy the same matrix system.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.