NAMING ELEMENTARY ROW OPERATIONS
and       PIVOTING
   
Need help doing row operations?
Try the Pivot Engine

An elementary row operation
is one of the following
changes made to the rows of
a matrix, as also described
in our text (Rolf Pg 89):
[1]
Interchanging two rows
[2]
Multiplying or dividing each element of a row
by the same non-zero number "c".
 
[3]
Adding to a row (the TARGET ROW) a non-zero
multiple "c" of another row (the TOOL ROW).
While pivoting (see below), the TOOL row is the PIVOT row.
 
 

PROF McFARLAND's NAMES FOR ROW OPERATIONS DIFFER A BIT from names used in our text: compare both in the table below. In the left column (Prof McFarland's) small r is used for rows in the (old) matrix about to be altered. Capital R is used to name the newly altered row in it's newly created matrix.

Prof McFarland's Names Row Operation Text's Names
[1] rm rn
or
rn = Rm and rm = Rn
Interchange
rows rm and rn
Rm Rn
[2] c. rm = Rm
or
. rm = Rm
Multiply
or
Divide		   by c
c. Rm Rm
or
. Rm Rm
[3] rm + c. rn = Rm
Add a Multiple
of Row n
to Row m		 Rm + c. Rn Rm

An example of how row operations should look is shown below, with all rows color coded (black/green/magenta/brown/blue) to show the match-up between matrix rows and their names.


3		   -1
2		 2
11		 r2 - 3r1 = R2
 1
0		   -1
 2
5		 (0.2)r2 = R2
 1
0		   -1
1		 2
1		 r1 + r2 = R1
 1
0		   0
1		 3
1

PIVOTING
Prof McFarland's names for row operations differ a bit from the names used in our text.

PIVOTING (or formally, a PIVOT TRANSFORMATION) uses row operations to change one matrix entry (the PIVOT) to "1" (see P1 below), and then (see P2 below) to change all other entries in the pivot's column into ZERO's. Also see our text (Rolf Pg 92).

Once a pivot is chosen, the row operations of pivoting MUST BE AS FOLLOWS:

P1
Change the chosen pivot into "1" by dividing the pivot's row by the pivot number, as was done in the 2nd row operation in the above example.
 
P2
Change the remainder of the pivot's COLUMN into 0's by adding to each row a suitable multiple of the PIVOT ROW, as was done in the 1st and 3rd row operations in the example above.

Note 1. The number changing to "1" is called a PIVOT, is usually encircled, and cannot be zero.

Note 2. Completely finish both P1 and P2 (above) before choosing another pivot.

Note 3. Prof McFarland has designed a time-saving interactive PIVOT ENGINE: check it out!

Note 4. (Again) Our brief example of pivoting, with row names color-matched with matrix rows:


3		   -1
2		 2
11		 r2 - 3r1 = R2
 1
0		   -1
 2
5		 (0.2)r2 = R2
 1
0		   -1
1		 2
1		 r1 + r2 = R1
 1
0		   0
1		 3
1

Thus, in the example just above, there are actually TWO pivot transformations shown. One consists of the 1st row operation alone, whose pivot was the number "1" in the original matrix, encircled; the other consists of the 2nd and 3rd row operations as a pair, whose pivot is the encircled number 5.

Note 5. Use pivoting when possible during Gauss/Jordan (Rolf, section 2.2) and when you find matrix inverses (Rolf, section 2.6). Though pivoting is not required to obtain correct answers in these sections, pivoting WILL be required later during simplex method, and now is a good time to develop the habits needed at those later times.