12.1 Elimination Procedure

To do Gaussian (or forward) elimination in this class, the general procedure consists of four basic steps. Take the initial submatrix to look at to be the complete given matrix. Then:

GE I

If there is only one row left in the submatrix currently being looked at, you are done with Gaussian elimination. And so you are if there are no more nonzero coefficients in the submatrix.

GE II

In the submatrix currently being looked at, identify the first column that still has a nonzero coefficient. Any earlier columns no longer appear anywhere, and should be dropped from the submatrix being looked at.

GE III

In the submatrix currently being looked at, use the first row to eliminate the first coefficient from the subsequent rows. Normally, you do that by subtracting suitable multiples of the first row from the subsequent ones. Step GE III can be described as “creating zeros below the pivot.” Note that this is only possible if the coefficient in the first column of the first row, called the pivot, is nonzero. In general, before doing the actual step GE III, you may need to exchange the current first row with a later one that will produce a better pivot. The next section will explain that in more detail.

GE IV

Drop the first row and the first column from the submatrix being looked at. With the so-reduced submatrix, repeat the process starting from step GE I.

This process will give rise to a so-called “echelon form” of the matrix. That is a special case of an upper triangular (if square) or upper trapezoidal (if not square) matrix.

Additional note: the eliminations in GE II can be done in a single step for all rows below the pivot. You do not have to show each row using a separate matrix. You must however explicitly show the multiplier(s) used.