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Gaussian Elimination2
45 �1 2 7
�2 6 9 0�7 5 �3 5
3
5Apply elementary row operations to the augmented matrix
2
64
5 �1 2 7
0 285
495
145
0 185 � 1
5745
3
75 (eqn 2)�✓�25
◆(eqn 1)
(eqn 3)�✓�75
◆(eqn 1)
2
64
5 �1 2 7
0 285
495
145
0 0 � 6510
655
3
75 (eqn 3)�
✓9
14
◆(eqn 2)
Solution : x1 = 3, x2 = 4, x3 = �2
PivotsMultipliers
1
Inverses of elimination matrices
E21
2
64
? ? ?
? ? ?
? ? ?
3
75
2
64
(row 1)
(row 2)� (`21)(row 1)
(row 3)
3
75 =
2
64
(row 1)
(row 2)
(row 3)
3
75
E�121
2
64
1 0 0
`21 1 0
0 0 1
3
75
2
64
(row 1)
(row 2)� (`21)(row 1)
(row 3)
3
75 =
2
64
(row 1)
(row 2)
(row 3)
3
75
(`21)(row 1) + (1) [(row 2)� (`21)(row 1)] = (row 2)
2
4X X XX X XX X X
3
5
2
4X X X0 X XX X X
3
5
2
64
1 0 0
�`21 1 0
0 0 1
3
75
2
64
(row 1)
(row 2)
(row 3)
3
75 =
2
64
(row 1)
(row 2)� (`21)(row 1)
(row 3)
3
75
Find
The inverse of the elimination matrix undoes the effect of elimination.
2
Inverses of elimination matrices
E31
(`31)(row 1) + (1) [(row 3)� (`31)(row 1)] = (row 3)
E�131
2
64
1 0 0
0 1 0
`31 0 1
3
75
2
664
(row 1)
(row 2)
(row 3)� (`31)(row 1)
3
775 =
2
64
(row 1)
(row 2)
(row 3)
3
75
2
64
1 0 0
0 1 0
�`31 0 1
3
75
2
64
(row 1)
(row 2)
(row 3)
3
75 =
2
64
(row 1)
(row 2)
(row 3)� (`31)(row 1)
3
75
2
4X X X0 X XX X X
3
5
2
4X X X0 X X0 X X
3
5
3
Inverses of elimination matrices
2
64
1 0 0
0 1 0
0 �`32 1
3
75
2
64
(row 1)
(row 2)
(row 3)
3
75 =
2
64
(row 1)
(row 2)
(row 3)� (`32)(row 2)
3
75
2
4X X X0 X X0 X X
3
52
4X X X0 X X0 0 X
3
5
upper triangular
E32
E�132
2
64
1 0 0
0 1 0
0 `32 1
3
75
2
64
(row 1)
(row 2)
(row 3)� (`32)(row 2)
3
75 =
2
64
(row 1)
(row 2)
(row 3)
3
75
(`32)(row 2) + (1) [(row 3)� (`32)(row 2)] = (row 3)
4
Inverse of elimination matrices
What is the product
=
2
64
1 0 0
`21 1 0
`31 `32 1
3
75
We can write A as the product of a lower triangular matrix and an upper triangular
matrix.
E32E31E21A = U
or
A = (E�121 E�1
31 E�132 )U = LU
The multipliers used in elimination appear in L.
(E32E31E21)�1 = E�1
21 E�131 E�1
32 ?
This is the LU decomposition of A.
E�121 E�1
31 E�132 =
2
64
1 0 0
`21 1 0
0 0 1
3
75
2
64
1 0 0
0 1 0
`31 0 1
3
75
2
64
1 0 0
0 1 0
0 `32 1
3
75
5
Practice!
A =
2
664
3 �7 �2 2�3 5 1 06 �4 0 �5
�9 5 �5 12
3
775
L =
2
664
1 0 0 0`21 1 0 0`31 `32 1 0`41 `42 `43 1
3
775 U =
2
664
X X X X0 X X X0 0 X X0 0 0 X
3
775
Numbers are all integers!
Check that you get LU = A.
Row reduce the matrix A to get an upper triangular matrix U . Along the way,
record the multipliers `ij you use in a lower triangular matrix L. Check that you
get LU = A.
The e
nd!
6
