The Gauss-Jordan method is an algebraic procedure used to solve systems of linear equations and compute matrix inverses. This method extends Gaussian elimination, transforming an augmented matrix into its row-reduced echelon form (RREF), where all nonzero rows start with a 1 and each of these "leading 1s" has zeros above and below in its column.
The process involves performing elementary row transformations on the matrix, such as:
- Swapping two rows.
- Multiplying a row by a nonzero scalar.
- Adding or subtracting a multiple of one row to another row.
Once the procedure is complete, the augmented matrix contains the solution to the system on its right-hand side, while the left-hand side will be an identity matrix if the system has a unique solution.