Recall from HELM booklet 8 that the basic idea with Gaussian (or Gauss) elimination is to replace the matrix of coefficients with a matrix that is easier to deal with. Usually the nicer matrix is of upper triangular form which allows us to find the solution by back substitution . For example, suppose we have
which we can abbreviate using an augmented matrix to
We use the boxed element to eliminate any non-zeros below it. This involves the following row operations
And the next step is to use the 2 to eliminate the non-zero below it . This requires the final row operation
This is the augmented form for an upper triangular system, writing the system in extended form we have
which is easy to solve from the bottom up, by back substitution .
Solve the system
The bottom equation implies that . The middle equation then gives us that
and finally, from the top equation,
Therefore the solution to the problem stated at the beginning of this Section is
The following Task will act as useful revision of the Gaussian elimination procedure.
Carry out row operations to reduce the matrix
into upper triangular form.
The row operations required to eliminate the non-zeros below the diagonal in the first column are as follows
Next we use the 5 on the diagonal to eliminate the 5 below it:
which is in the required upper triangular form.