Solve Gauss Jordan Elimination Calculator

Gauss-Jordan Elimination Method:

\[ [A|b] \rightarrow [I|x] \]

Where A is the coefficient matrix, b is the constant vector, I is the identity matrix, and x is the solution vector.

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1. What is Gauss-Jordan Elimination?

Gauss-Jordan elimination is an algorithm for solving systems of linear equations. It transforms the augmented matrix [A|b] into reduced row echelon form to find the solution vector x.

2. How the Calculator Works

The calculator performs the following steps:

Creates an augmented matrix [A|b] from your input
Uses partial pivoting to ensure numerical stability
Performs row operations to achieve reduced row echelon form
Extracts the solution from the rightmost column

3. Applications

Details: Gauss-Jordan elimination is used in engineering, physics, computer graphics, economics, and many other fields that require solving systems of linear equations.

4. Using the Calculator

Tips:

Select the matrix size (2x2, 3x3, or 4x4)
Enter the coefficients of matrix A and vector b
Click "Calculate" to see the solution
For best results, avoid matrices with determinant close to zero

5. Frequently Asked Questions (FAQ)

Q1: What if my matrix is singular?
A: The calculator will notify you if the matrix is singular (no unique solution exists).

Q2: How accurate are the results?
A: Results are accurate to 6 decimal places, though rounding errors may occur with ill-conditioned matrices.

Q3: Can I solve larger systems?
A: This calculator handles up to 4x4 systems. For larger systems, specialized software is recommended.

Q4: What's the difference between Gaussian and Gauss-Jordan elimination?
A: Gaussian elimination produces row echelon form, while Gauss-Jordan continues to reduced row echelon form.

Q5: Why use partial pivoting?
A: Partial pivoting improves numerical stability by reducing rounding errors.