2.4.4 Gaussian-Jordan Elimination

Gauss-Jordan Elimination is a variant of Gaussian Elimination. Again, we are transforming the coefficient matrix into another matrix that is much easier to solve, and the system represented by the new augmented matrix has the same solution set as the original system of linear equations. In Gauss-Jordan Elimination, the goal is to transform the coefficient matrix into a diagonal matrix, and the zeros are introduced into the matrix one column at a time. We work to eliminate the elements both above and below the diagonal element of a given column in one pass through the matrix.

The general procedure for Gauss-Jordan Elimination can be summarized in the following steps:

An Example.

We will apply Gauss-Jordan Elimination to the same example that was used to demonstrate Gaussian Elimination. Remember, in Gauss-Jordan Elimination we want to introduce zeros both below and above the diagonal.

1. Write the augmented matrix for the system of linear equations.

As before, we use the symbol to indicate that the matrix preceding the arrow is being changed due to the specified operation; the matrix following the arrow displays the result of that change.

2. Use elementary row operations on the augmented matrix [A|b] to transform A into diagonal form.

At this point we have a diagonal coefficient matrix. The final step in Gauss-Jordan Elimination is to make each diagonal element equal to one. To do this, we divide each row of the augmented matrix by the diagonal element in that row.

3. By dividing the diagonal element and the right-hand-side element in each row by the diagonal element in that row, make each diagonal element equal to one.

When performing calculations by hand, many individuals choose Gauss-Jordan Elimination over Gaussian Elimination because it avoids the need for back substitution. However, we will show later that Gauss-Jordan elimination involves slightly more work than does Gaussian elimination, and thus it is not the method of choice for solving systems of linear equations on a computer.