Understanding Gaussian Elimination Method in Linear Algebra

 
Mustansiriyah University
College of Engineering
Computer Engineering Dept
 
Mathematical analysis II
 
G
AUSS
 
J
O
R
DAN
 
ELIMINATION
METH
O
D
Lect. Sarmad K. Ibrahim
 
G
AUSS
 
J
O
R
DAN
 
METH
O
D
 
Some
 
a
u
th
o
rs
 
use
 
the
 
t
e
rm
 
Ga
u
ssian
eli
m
ination
 
to
 
ref
e
r
 
o
n
ly
 
to
 
the
 
pr
o
ce
d
ure
 
u
n
til
 
the
matrix
 
is
 
in
 
e
ch
e
lon
 
form,
 
a
n
d
 
use
 
t
h
e
t
e
r
m
 
G
a
u
s
s
-
J
o
r
d
a
n
 
e
l
i
m
i
n
a
t
i
o
n
 
t
o
 
r
e
f
e
r
 
t
o
 
t
h
e
p
r
o
c
e
d
u
r
e
 
w
h
i
c
h
 
e
n
d
s
 
i
n
 
r
e
d
u
c
e
d
 
e
c
h
e
l
o
n
 
f
o
r
m
.
 
 
In
 
line
a
r
 
alg
e
bra,
 
Ga
u
s
s
Jord
a
n
 
e
lim
i
n
a
tion
 
is
 
an
alg
o
rithm
 
for
 
g
e
tting
 
matrices
 
in
 
re
d
uc
e
d
 
r
ow
echel
o
n
 
form
 
using
 
ele
m
ent
a
ry
 
r
ow
 
o
p
era
t
io
n
s.
 
It
is
 
a
 
v
a
riation
 
of
 
Ga
u
ssian
 
eliminatio
n
.
 
H
IS
T
O
R
Y
 
ABOUT
 
G
AUSS
 
J
ORDAN
METHOD
 
  
 
it
 
is
 
a
 
v
a
riation
 
of
 
Ga
u
ssian
 
elimination
d
e
scribed
 
by
 
 
Wi
l
h
e
lm
 
Jordan
 
in
 
1
8
8
7.
 
as
 
 
Howev
e
r
,
 
t
h
e
 
me
t
h
o
d
 
a
lso
 
a
p
p
e
a
rs
 
in
 
a
n
 
article
by
 
Clas
e
n
 
publish
e
d
 
in
 
t
h
e
 
s
a
me
 
ye
a
r
.
 
J
o
rdan
a
n
d
 
C
lasen
 
pro
b
a
b
ly
 
discov
e
red
 
Gaus
s
Jord
a
n
eli
m
ination
 
inde
p
en
d
entl
y
.
 
 
H
ERE ARE
 
THE
 
STE
P
S
 
T
O
 
G
AUS
S
-
J
ORD
A
N
ELIMIN
A
TION
:
 
 
T
urn
 
the eq
u
ati
o
ns
 
i
n
to an au
g
mented
 
matri
x
.
 
 
Use e
l
ementary
 
row operations
 
on mat
r
ix
 
[
A
|
b]
 
to
 
t
r
ansform
 
A
 
i
nto
 
d
i
ag
o
nal
 
fo
r
m.
 
Make s
u
re 
t
here are
no zeros in the d
i
a
g
o
n
a
l
.
 
 
D
i
vi
d
e
 
the diag
o
nal
 
e
l
ement
 
and
 
the r
i
gh
t
-
hand
e
l
eme
n
t
 
(of
 
b) 
f
or
 
that 
d
i
a
g
o
n
a
l
 
e
l
eme
n
t's
 
row
 
so
that the
 
d
i
ag
o
nal
 
e
l
ement
 
is equ
a
l
 
to 
o
ne.
 
 
E
x
a
m
p
l
e
 
1
.
 
S
o
l
v
e
 
t
h
e
 
f
o
l
l
o
w
i
n
g
 
s
y
s
t
e
m
 
b
y
 
u
s
i
n
g
 
t
h
e
 
Ga
u
s
s
-
J
o
r
d
an
 
e
l
imi
n
at
i
on
 
met
h
o
d
.
 
x +
 
y +
2x +
 
3y
4x + 
5
z
 
z =
 
5
+ 5z = 8
= 2
 
 
S
o
l
u
t
i
o
n
:
fo
l
l
o
win
g
.
The
augmented
m
atrix
 
of
the
system
is
the
1
2
4
1
3
0
1
|5
5
|8
5
|2
 
 
 
W
e
 
w
i
ll
 
now
 
p
erf
o
rm
 
row
 
op
e
rat
i
ons
 
u
n
til
 
we
 
o
bt
a
in
 
a
 
matr
i
x
 
in
 
r
e
du
c
ed
 
row
 
ec
h
el
o
n
 
form.
 
1
2
4
 
1
3
0
 
1
|5
5
|8
5
|2
 
0
 
1
 
3
 
-2
R
3-4
R
1
 
0
 
-4
 
1
 
-18
R
3
+4R2
 
0
 
0
 
13
 
-26
(1/
1
3)
R
3
 
0
 
0
 
1
 
-2
R2-3R3
 
0
 
1
 
0
 
4
 
1
 
1
 
0
 
7
R
1-
R
3
 
1
 
0
 
0
 
3
R1
-R2
 
 
F
r
om
 
t
hi
s
 
fi
n
a
l
 
mat
ri
x,
s
yst
e
m.
 
It
 
is
we
c
an
 
r
e
a
d
t
h
e
 
s
o
l
u
ti
on
of
 
t
he
X= 3
Y=4
Z=-2
 
 
Substitute x , y, z in system equation, if the right side = left
side in three equation then the sol. is correct
 
3+4-2=5
2*3+3*4+5*(-2)=8
4*3+5*(-2)=2
 
Thank you
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Gaussian Elimination and Gauss-Jordan Elimination are methods used in linear algebra to transform matrices into reduced row echelon form. Wilhelm Jordan and Clasen independently described Gauss-Jordan elimination in 1887. The process involves converting equations into augmented matrices, performing row operations to simplify the matrix, and obtaining a reduced row echelon form. An example system solving process is also demonstrated step by step.


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  1. Mustansiriyah University College of Engineering Computer Engineering Dept Mathematical analysis II GAUSS JORDAN ELIMINATION METHOD Lect. Sarmad K. Ibrahim

  2. GAUSS JORDAN METHOD Some authors use the term Gaussian elimination to refer only to the procedure until the matrix is in echelon form, and use the term Gauss-Jordan elimination to refer to the procedure which ends in reduced echelon form. In linear algebra, Gauss Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations. It is a variation of Gaussian elimination.

  3. HISTORYABOUT GAUSS JORDAN METHOD it is a variation of Gaussian elimination described by Wilhelm Jordan in 1887. as However, the method also appears in an article by Clasen published in the same year. Jordan and Clasen probably discovered Gauss Jordan elimination independently.

  4. HERE ARE THE STEPS TO GAUSS-JORDAN ELIMINATION: Turn the equations into an augmented matrix. Use elementary row operations on matrix [A|b] to transformAinto diagonal form. Make sure there are no zeros in the diagonal. Divide the diagonal element and the right-hand element (of b) for that diagonal element's row so that the diagonal element is equal to one.

  5. Example 1 . Solve the following system by using the Gauss-Jordan elimination method. x + y + 2x + 3y 4x + 5z z = 5 + 5z = 8 = 2

  6. Solution: following. The augmented matrix of the system is the 1 2 4 1|5 5|8 5|2 1 3 0 We will now perform row operations until we obtain a matrix in reduced row echelon form. 5 1 1 1 1 2 4 1 3 0 1|5 5|8 5|2 -2 0 1 3 2 4 0 5

  7. 1 1 1 5 1 1 1 5 0 1 3 -2 R3-4R1 0 1 3 -2 0 -4 -18 1 4 0 5 2

  8. 1 1 1 5 1 1 1 5 R3+4R2 0 1 3 -2 0 1 3 -2 0 -26 0 0 -4 1 -18 13

  9. 1 1 1 5 1 1 1 5 (1/13)R3 0 1 3 -2 0 1 3 -2 0 0 13 -26 0 0 -2 1

  10. 1 1 1 5 1 1 1 5 0 1 3 -2 0 1 4 0 R2-3R3 0 0 1 -2 0 0 1 -2

  11. 1 1 1 5 1 1 0 7 0 1 0 4 0 1 0 4 R1-R3 0 0 1 -2 0 0 1 -2

  12. 1 1 0 7 1 0 0 3 0 1 0 4 0 1 0 4 R1-R2 0 0 1 -2 0 0 1 -2

  13. From this final matrix, system. It is we can read the solution of the X= 3 Y=4 Z=-2 Substitute x , y, z in system equation, if the right side = left side in three equation then the sol. is correct 3+4-2=5 2*3+3*4+5*(-2)=8 4*3+5*(-2)=2

  14. Thank you

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