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Linear Systems and Matrix Formalism
Adam Gamzon
This handout will examine rowreduction and GaussJordan elimination in more detail with
a heavy emphasis on examples. In class, we introduced matrices as a way of encoding all of
the information in a given linear system in order to save on the bookkeeping necessary while
performing the algorithm for GaussJordan elimination. To recall how this goes, let’s look at an
example.
Example 1.
Consider the system
2
x
1
+4
x
2
+3
x
3
+5
x
4
=3
4
x
1
+8
x
2
+7
x
3
x
4
=4

2
x
1

4
x
2
x
3
x
4
=0
x
1
+2
x
2
x
3

x
4
=2
.
The
coe
f
cient matrix
for this system is
0
B
B
@
24
3
5
48
7
5

2

43 4
12
2

1
1
C
C
A
.
As mentioned in class, the coe
f
cient matrix will be less important for us early in the course
but it will come in handy in a couple of chapters.
The
augmented matrix
for this system is
A
=
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