5.3. Gaussian and Gauss-Jordan Elimination

Gaussian and Gauss-Jordan Elimination are methods to bring a matrix to row echelon and reduced row echelon form, respectively. Row echelon form (often abbreviated REF) is often defined by the first three of the following rules while reduced row echelon form (RREF) is defined by all four:

All zero rows are at the bottom of the matrix.
If a pivot is defined as the first non-zero entry of any given row, then the pivot in each row after the first occurs at least 1 column further to the right than the previous row.
The pivot in any nonzero row is 1.
All entries in the column above and below a pivot are zero.

Some texts omit rule 3 from the REF definition so make sure to consult your actual course text for the definition you will use. In any case, the definition of reduced row echelon form (RREF) always contains the first 3 rules and the 4th rule.

For example, the following matrices are an example of a regular matrix and its corresponding REF form:

Note that the ones along the diagonal are the pivots. A matrix need not be square in order to have an REF form:

Note that again here the pivots are all 1 so this matrix is in REF form, but the entries in the columns above and below the pivots are not zero (Rule #4) so this matrix is not in RREF form.

Excercise 5-3.

Identify each of the matrices as being either in REF, RREF, or neither. If the matrix is not in REF or RREF form, state the rule(s) it violates.

This matrix is not in REF or RREF form because all the zero rows should be at the bottom (Rule #1).
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This matrix is in REF form but not RREF form because the in the top row needs to be a since it is in the same column as a pivot (Rule #4).
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This matrix is in RREF form. Even though it has an 8 in the top row, the second column is not pivotal since only the first and third contain pivots.
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This matrix is not in REF or RREF form. The second and third rows need to be swapped in order to get this matrix to RREF form. The pivots have to occur from left to right when moving down each row (Rule #2).
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This matrix is not in REF or RREF form. For REF, the pivots in the last two rows need to be equal to 1 (Rule #3). For RREF the entry in the second row, third column needs to be 0 (Rule #4).
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