Elimination of Variables.
Example: Solve the system of equations 3x + 2y =7
4x + 3y = 9
To eliminate x, we need the coefficients of x to have the same magnitude but opposite
sign. Multiply the first equation by 4 and the second by -3. The result is
12x + 8y = 28
-12x - 9y = -27
Adding these equations gives
-y = 1
so y = -1.
Then substituting y = -1 into one of the original equations gives x = 3.
In Gauss-Jordan, we use elimination of variables in larger systems to eliminate all the
variables we can. We also divide each nonzero row by the leading nonzero coefficient so
the first coefficient is always 1.
Gauss-Jordan in matrices:
The only allowed row operations are:
1) Multiply a row by a nonzero number. Denote with aRi which means row i is
multiplied by the nonzero number a.
2) Swap places of rows. Denote swapping rowi and rowj as Ri  Rj. This is used if the
pivot position has a 0 entry.
3) Add a multiple of one row to another row. Denote with Ri + aRj which means a times
row j is added to rowi. Note the row being replaced is always mentioned first.
Definitions: 1) A unit column is a column with one 1 and all other entries are 0.
2) An entry in a row is a leading 1 if it is one and it is the first nonzero entry
in that row.
3) To pivot means to use the allowed row operations to create a unit column
with the one in the specified position.
The first pivot is done on row1 column1 if that entry is not zero. If it is 0, swap rows to
get a nonzero entry. If the whole first column is 0, move to the 2nd column.
You make a unit column in column 1, then pivot in row2 column 2, if that entry is not 0.
Keep moving over and down until all pivots are done.
the result of all pivots is called the reduced row echelon form or rref. (echelon means
stairstepped)
A matrix is in rref when:
1) All rows of only 0's are at the bottom.
2) Each nonzero row begins with a leading 1.
3) If a column contains a leading 1 then it is a unit column.
4) The leading ones are arranged from upper left to lower right. (in echelon form)
Standard Pivot Procedure assuming the pivot entry is not 0.
1) Divide the pivot row by the pivot entry.
2) Add a multiple of the pivot row to another row to get 0 in the pivot column of that
row.
3) Repeat step 2 until the pivot column is a unit column.
Example:
 2

5


 3
6
3
2
 4

1

6 

To pivot on row1 column1, first divide row 1 by 2. Then add -5
times the new row 1 to row2. Then add 3 times the same new row 1 to row 3.
This would be denoted
(1/2)R1, R2+(-5)R1, R3+3R1.
The result is the matrix:
1

0


0
3
12
11
 2

11

0 

The next pivot position is row 2 column 2.