An Example of
Gauss Elimination
Original matrix:
An Example of
Gauss Elimination
4
4
8
2
8
7
1
3
6
Original matrix:
I
4
4
8
II
2
8
7
III
1
3
6
4
4
8
-1/2 I + II:
0
6
3
-1/4 I + III:
0
2
4
Step 1: Put zeros in
first column
1999,2001
Dianne P. O'Leary
1
An Example of
Gauss Elimination
Current matrix:
Dianne P. O'Leary
2
What Gauss Elimination
Accomplished:
I
4
4
8
4
4
8
4
4
8
II
0
6
3
2
8
7
0
6
3
III
0
2
4
1
3
6
0
0
3
4
4
8
0
6
3
0
0
3
Step 2: Put zeros in
second column
-1/3 II + III:
1999,2001
1999,2001
Dianne P. O'Leary
3
1999,2001
Dianne P. O'Leary
4
1
But Notice: The multipliers
are important, too:
4
4
8
2
8
7
1
3
6
=
But Notice: The multipliers
are important, too:
1
0
0
4
4
8
4
4
8
1/2
1
0
0
6
3
2
8
7
1
0
0
3
1
3
6
1/4 1/3
=
Dianne P. O'Leary
5
How this solves linear
systems:
4
4
8
x1
2
8
7
x2
1
3
6
x3
=
4
8
1/2
1
0
0
6
3
1
0
0
3
L
U
Dianne P. O'Leary
6
Ly=b
1
0
0
y1
1/2
1
0
y2
1
y3
5
5
1/4 1/3
Step 1:
Solve L y = b
LU x = b
Step 2:
Solve U x = y
Dianne P. O'Leary
4
4
Ax=b
1999,2001
0
=
1999,2001
Solve
0
1/4 1/3
A
1999,2001
1
7
1999,2001
Dianne P. O'Leary
4
=
5
5
8
2
Solve
Ly=b
1
0
0
y1
1/2
1
0
y2
1
y3
1/4 1/3
1999,2001
Solve
Solve
=
4
1
0
0
y1
5
1/2
1
0
y2
5
1/4 1/3
1
y3
4
=
1/2 y1 + y2 = 5,
y1 = 4
y2 = 3
9
Ly=b
1999,2001
0
0
y1
1/2
1
0
y2
1
y3
1/4 y1 + 1/3 y2 +y3 = 5,
=
y1 = 4
Dianne P. O'Leary
Result:
1
5
5
1 y1 = 4
Dianne P. O'Leary
1/4 1/3
Ly=b
10
Ly=b
4
1
0
0
4
5
1/2
1
0
3
5
1/4 1/3
1
3
4
=
5
5
y1 = 4, y2 = 3
y3 = 3
1999,2001
Dianne P. O'Leary
11
1999,2001
Dianne P. O'Leary
12
3
Now solve
Ux=y
4
4
8
x1
0
6
3
x2
0
0
3
x3
=
Solve
Ux=y
4
4
4
8
x1
3
0
6
3
x2
3
0
0
3
x3
4
=
3
3
3 x3 = 3
x3 = 1
1999,2001
Solve
Dianne P. O'Leary
13
Ux=y
1999,2001
Solve
4
4
8
x1
0
6
3
x2
0
0
3
x3
=
4
4
8
x1
3
0
6
3
x2
3
0
0
3
x3
4
=
3
3
4 x1 + 4 x2 + 8 x3 = 4, x3 = 1, x2=0
x1 = -1
x2 = 0
Dianne P. O'Leary
14
Ux=y
4
6 x2 + 3 x3 = 3, x3 = 1
1999,2001
Dianne P. O'Leary
15
1999,2001
Dianne P. O'Leary
16
4
Result:
1999,2001
Ux=y
4
4
8
-1
0
6
3
0
0
0
3
1
Dianne P. O'Leary
Result:
Ax=b
4
4
8
-1
2
8
7
0
1
3
6
1
4
=
4
=
5
3
5
3
17
1999,2001
Dianne P. O'Leary
18
5