Gauss-Jordan method

The Gauss-Jordan method is an algorithm used to solve systems of linear equations and to find the inverse of a square matrix. It is an extension of the Gaussian elimination method, which is used to convert a matrix into row-echelon form. The Gauss-Jordan method further transforms the matrix into reduced row-echelon form, making it particularly useful for finding the solutions to systems of linear equations and computing matrix inverses.

Here are the steps of the Gauss-Jordan method:

- Step 1: Augmented Matrix

Write the system of linear equations as an augmented matrix, which combines the coefficient matrix and the constant terms on the right-hand side.

- Step 2: Row Operations

Perform row operations to convert the augmented matrix into row-echelon form. The row-echelon form has the following properties:

a) The first nonzero element in each row (known as the leading entry) is 1.

b) The leading 1 in each row is to the right of the leading 1 in the row above it.

c) All entries below and above the leading 1's are zero.

   To achieve the row-echelon form, use elementary row operations, which include:

a) Swapping two rows.

b) Multiplying a row by a nonzero constant.

c) Adding a multiple of one row to another row.

- Step 3: Backward Elimination

Starting from the bottom row of the matrix, perform backward elimination to create zeros above each leading 1. This process involves using elementary row operations to subtract multiples of rows from each other, aiming to create zeros above the leading 1's.

- Step 4: Reduced Row-Echelon Form

Continue with the backward elimination until the entire matrix is in reduced row-echelon form. The reduced row-echelon form has the same properties as the row-echelon form, with the additional requirement that each leading 1 is the only nonzero entry in its column.

- Step 5: Solutions and Inverse

Once the matrix is in reduced row-echelon form, you can read the solutions to the system of linear equations directly from the rightmost column of the augmented matrix. If you were working with a square matrix to find its inverse, the inverse will be found on the right side of the vertical bar, with an identity matrix on the left side.

If the reduced row-echelon form has rows with all zeros, indicating inconsistent equations (no solution) or if there are rows with leading 1's in the last column, it indicates an underdetermined system with infinitely many solutions.

The Gauss-Jordan method is efficient and numerically stable for small to moderate-sized matrices. However, for large matrices, other algorithms like LU decomposition or iterative methods may be more practical.