Matrix Elimination Method Calculator

Enter matrix data

Choose the system size, fill the coefficient matrix and constants, then solve using matrix elimination with partial pivoting.

Matrix size

Display precision

Elimination mode

a11

a12

a13

b1 constant

a21

a22

a23

b2 constant

a31

a32

a33

b3 constant

Reset calculator

Formula used

For a linear system A x = b, the calculator builds the augmented matrix [A|b] and applies row elimination until it reaches upper triangular form.

Row operation: Rj = Rj - (a_j,i / a_i,i) Ri

Determinant after elimination: det(A) = (-1)^s × Π u_ii, where s is the number of row swaps.

Back substitution: x_i = (u_i,n+1 - Σ u_ijx_j) / u_ii

Consistency test: compare rank(A) and rank([A|b]). Equal full ranks give a unique solution. Equal lower ranks give infinitely many solutions. Different ranks mean no solution.

How to use this calculator

Select a matrix size from 2 × 2 up to 5 × 5.
Enter each coefficient into the matrix input fields.
Enter the constant terms for the right-hand side vector.
Choose the display precision you prefer.
Click Solve system to run the elimination method.
Review the status, determinant, ranks, solution vector, and residual checks.
Scroll through the elimination steps to verify every row operation.
Download a CSV or PDF report for revision, classwork, or records.

Example data table

This example corresponds to the default 3 × 3 sample system.

Equation	Coefficient row	Constant
2x + y - z = 8	[2, 1, -1]	8
-3x - y + 2z = -11	[-3, -1, 2]	-11
-2x + y + 2z = -3	[-2, 1, 2]	-3
Expected solution	[x, y, z]	[2, 3, -1]

FAQs

1. What does this calculator solve?

It solves square systems of simultaneous linear equations by converting the augmented matrix into an upper triangular form, then applying back substitution when a unique solution exists.

2. Why does the calculator use partial pivoting?

Partial pivoting swaps rows to place a larger pivot in position. That reduces division by tiny numbers, improves numerical stability, and helps the elimination process stay reliable.

3. What does rank tell me?

Rank shows how many independent rows remain after elimination. It helps classify the system as uniquely solvable, inconsistent, or having infinitely many solutions.

4. Why can the determinant become zero?

A zero determinant means the coefficient matrix is singular. In that case, the system cannot have a single unique solution, and it may instead be inconsistent or underdetermined.

5. What are residual checks?

Residuals compare the computed left-hand side, Ax, with the original constants b. Small residuals show that the solution satisfies the equations closely.

6. Can I use decimals or negative values?

Yes. The input fields accept integers, decimals, and negative numbers. That makes the tool suitable for classroom examples, engineering-style systems, and homework practice.

7. What if the calculator says infinite solutions?

That means some equations depend on others, so the system has free variables. The rows reduce to a dependent form rather than giving one fixed answer set.

8. Why export the result to CSV or PDF?

Exports make it easier to save solved systems, document teaching examples, compare results, and share a clean summary with students, classmates, or colleagues.