Describe your change:

This pull request adds a complete implementation of the Gauss–Jordan elimination algorithm for solving systems of linear equations of the form Ax= b . The function reduces the coefficient matrix to Reduced Row Echelon Form (RREF) while applying the same transformations to the right-hand-side vector.

Key highlights of the change:

1. Implements a robust algorithm with pivot detection, row swaps, normalization, and elimination

2. Handles both consistent and singular systems gracefully

3. Performs validation to ensure input shape compatibility

4. Includes doctests that verify correctness.

5. Written to be readable and educational, aligning with the goals of TheAlgorithms/Python repository.

This function is a core algorithm in linear algebra and complements existing matrix utilities in the repository.

• Add an algorithm?
• Fix a bug or typo in an existing algorithm?
• Add or change doctests? -- Note: Please avoid changing both code and tests in a single pull request.
• Documentation change?

Checklist:

• I have read CONTRIBUTING.md.
• This pull request is all my own work -- I have not plagiarized.
• I know that pull requests will not be merged if they fail the automated tests.
• This PR only changes one algorithm file. To ease review, please open separate PRs for separate algorithms.
• All new Python files are placed inside an existing directory.
• All filenames are in all lowercase characters with no spaces or dashes.
• All functions and variable names follow Python naming conventions.
• All function parameters and return values are annotated with Python type hints.
• All functions have doctests that pass the automated testing.
• All new algorithms include at least one URL that points to Wikipedia or another similar explanation.
• If this pull request resolves one or more open issues then the description above includes the issue number(s) with a closing keyword: "Fixes #ISSUE-NUMBER".