Matrix Representation of a System of Equations
A matrix is a rectangular table of numbers. We can write a list of linear expressions in
by putting
the equations in the same order with first and second in each equation, and then putting only the
coefficients from each equation into the matrix.
3
Example 1:
becomes the matrix 
4
To represent the system of equations
3
2
4
3
follows: 
2

3
we use an augmented matrix as
7

9
Example 2:
Realign the system
 3
1
 4
1
The augmented matrix form is: 
5 

 9
We want to reduce this system to one that has the same solution set so that we can read the solution.
If the solution is unique, we can reduce it to the system,
The augmented matrix then would be:
1

0
0
1
.
a

b
The procedure is to do a sequence of pivots until we have this augmented matrix.
What is a pivot? A pivot is a sequence of row operations that reduces a column to a unit column. A unit
column is a column with one 1 and all other entries 0. This means we have eliminated that variable from
all but one equation.
For example 2, the first pivot is
 3

 4
1
1

5 
1

 (1/3) R1 -
 9
 4

1
3
1

5 
1

3  R2+4R1- 
 9 
0

Now the first column is a unit column. The pivot is done.
1
3
7
3
5 
3 
7
 
3
Now we need to do the 2nd pivot and eliminate

1
(3/7)R2 
0

1
3
1
5 
1
3  R1- (1/3)R2 
0
 1 
from the first equation:
0
1
2 

 1
And we read the solution: x  2 , y   1
When no more pivots can be done, the augmented matrix is in rref.
What is RREF? Gauss Jordan process
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Ri + aRj which means Ri is replaced by
Ri plus aRj
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)
Now we need to consider the possibilities of no solution and infinitely many solutions.
If there is no solution, the reduced matrix of 1’s and 0’s has an inconsistent row, that is a row that says
0 is a nonzero number, such as 0 0 | 1 or 0 0 |3, etc.
If there are infinitely many solutions, at least one of the variables cannot be eliminated.
1

A 3x3 system in which this happens could have an rref that looks like: 0

 0
1

0
0
2
1
3
5 
1
or


 6
0
2
1
0
0
0
2
1
3
0
0
5 

 6 or

0 
5

0
What would a 2x2 with infinitely many solutions look like?
Remember, a row of 0’s only means one equation is redundant. It does not guarantee infinitely many
solutions.
Examples:
1.
Each augmented matrix represents a system of equations in
a)
1

0

 0
0
0
1
0
0
1
1 

3

4 
b)
1

0
1
2
1
3
Find the solution set.
4 

 3
c)
1

0

0

0
0
2
1
3
0
0
0
0
2. Write the system as an augmented matrix and perform the next pivot.
x  2 y  5 z  3,
3 y  z  0,
 4y  z  7
3. Determine whether or not the matrix is in rref.
a)
0

1

 0
1
0
0
0
0
2
2

3

5 
b)
0

1
1
2
0
1
5

3
c)
1

0

 0
0
0
2
1
0
3
7

5

6 
4

5

2

0