Coefficient Matrix Solver
Enter the coefficient matrix A and the right-hand vector b for a system written as A x = b.
Example Data Table
This sample 3 × 3 system produces the unique solution x = [2, 3, -1].
| Equation | a1 | a2 | a3 | Right-Hand Value |
|---|---|---|---|---|
| Row 1 | 2 | 1 | -1 | 8 |
| Row 2 | -3 | -1 | 2 | -11 |
| Row 3 | -2 | 1 | 2 | -3 |
Formula Used
System Form
A x = b, where A is the coefficient matrix, x is the unknown vector, and b is the right-hand vector.
Determinant Rule
If det(A) ≠ 0, the matrix is nonsingular and the system has one unique solution.
Rank Test
If rank(A) = rank([A|b]) = n, the solution is unique. If the common rank is smaller than n, infinitely many solutions exist. If rank(A) < rank([A|b]), the system is inconsistent.
Inverse Method
When A is invertible, the solver also uses x = A-1b as a consistency check after Gaussian elimination.
Residual Check
The residual vector is r = Ax - b. Values near zero indicate a numerically consistent answer.
How to Use This Calculator
- Select the matrix size from 2 × 2 up to 5 × 5.
- Enter every coefficient in matrix A.
- Enter the matching right-hand values in vector b.
- Choose the decimal precision for displayed results.
- Set a pivot tolerance when you need tighter numerical control.
- Click Solve Matrix System to calculate the determinant, ranks, solution type, and solution details.
- Download the resulting report as CSV or PDF when needed.
Frequently Asked Questions
1. What does a coefficient matrix represent?
A coefficient matrix stores the numerical multipliers of all unknown variables in a linear system. It separates the system structure from the right-hand constants, making matrix operations possible.
2. Why is the determinant important here?
The determinant quickly indicates whether a square coefficient matrix is singular. A nonzero determinant usually means the system has one unique solution and an inverse exists.
3. What is the difference between rank(A) and rank([A|b])?
Rank(A) measures independent rows in the coefficient matrix. Rank([A|b]) measures independence after appending the constants. Comparing them reveals whether the system is consistent or inconsistent.
4. When do infinitely many solutions occur?
Infinitely many solutions occur when the system is consistent but dependent. In practice, rank(A) equals rank([A|b]) but both are smaller than the number of unknowns.
5. Why does the solver show a residual vector?
The residual vector checks the computed solution numerically. If each residual entry is very close to zero, the displayed answer satisfies the original equations within tolerance.
6. What does pivot tolerance control?
Pivot tolerance defines when a very small number should be treated as zero during elimination. It helps stabilize results for nearly singular or poorly scaled matrices.
7. Can this page solve inconsistent systems?
Yes. The page classifies inconsistent systems by comparing matrix ranks. It reports that no solution exists when the augmented system introduces an impossible equation.
8. Is the inverse always available?
No. The inverse appears only when the coefficient matrix is nonsingular. Singular matrices have zero determinant and cannot produce a standard inverse matrix.