Page 44 - 35Linear Algebra

44 Systems of Linear Equations

The following properties define RREF:

1. In every row the left most non-zero entry is 1 (and is called a pivot).

2. The pivot of any given row is always to the right of the pivot of the
row above it.

3. The pivot is the only non-zero entry in its column.

Example 16 (Augmented matrix in RREF)

 
1 0 7 0
 0 1 3 0 
 
0 0 0 1
 
0 0 0 0
Example 17 (Augmented matrix NOT in RREF)

 
1 0 3 0
0 0 2 0
 
 
 0 1 0 1 
0 0 0 1
Actually, this NON-example breaks all three of the rules!
The reason we need the asterisks in the general form of RREF is that
not every column need have a pivot, as demonstrated in examples 13 and 16.
Here is an example where multiple columns have no pivot:

Example 18 (Consecutive columns with no pivot in RREF)

x + y + z + 0w = 2 1 1 1 0 2 1 1 1 0 2
⇔ ∼
2x + 2y + 2z + 2w = 4 2 2 2 1 4 0 0 0 1 0

x + y + z = 2
⇔
w = 0 .
Note that there was no hope of reaching the identity matrix, because of the shape of
the augmented matrix we started with.

With some practice, elimination can go quickly. Here is an expert showing
you some tricks. If you can’t follow him now then come back when you have
some more experience and watch again. You are going to need to get really
good at this!

44