Elimination using matrices
Recall : Multiplication on the left by a row vector results
in a row vector.
⇥
a
a
⇥
b
2
c
4
⇤ 2
2
1
2
⇤
⇥
4
5
+b
3
4
9
1
1
5
3 5=
7
9
3
⇤
Result is a row vector
+c
⇥
2
1 7
⇤
Multiplication on the left can be thought of as a linear
combination of rows.
Monday, October 7, 13
1
Elimination using matrices
⇥
a
b
(a, b, c) = ?
c
⇤ 2
4
4
9
1
(0, 1, 0)
What about
⇥
⇤ 2
a b c
4
(a, b, c) = (1, 0, 1)
Monday, October 7, 13
2
1
2
2
1
2
4
9
1
3
5
⇥
3 5= 1 9
7
3
⇤
3
5
⇥
⇤
3 5 = 0 5 12
7
We managed to
eliminate the first
coefficient
2
Elimination matrices
2x + 4y
2z = 2
4x + 9y
3z = 8
2x
2
4
elimination
2x + 4y
2z = 2
y+z =4
3y + 7z = 10
2
4
2
4
9
3
A
3
2
3 5
7
4z = 8
matrix
multiplication?
E
2
2
4 0
0
4
1
0
U
3
2
1 5
4
Find a matrix E so that EA = U
Monday, October 7, 13
3
Elimination matrices : First step
2x + 4y
4x + 9y
2x
eliminate
2
1
4 ?
?
2z = 2
2x + 4y
elimination
3z = 8
y+z =4
3y + 7z = 10
0
?
?
32
0
? 54
?
2z = 2
y + 5z = 12
2
4
2
4
9
3
3
2
2
2 4
3 5=4 0 1
7
0 1
3
2
1 5
5
How do we express ”(equation 2) (2)(equation 1)”
How do we express ”(equation 3) ( 1)(equation 1)”
Monday, October 7, 13
4
Elimination matrices : First step
How do we express ”(equation 2) (2)(equation 1)”
2
4
`21
1
2
?
32
0 0
1 0 54
? ?
2
4
2
4
9
3
How do we express ”(equation
2
32
1 0 0
2
4
4 2 1 0 54 4
9
1 0 1
2
3
`31
Monday, October 7, 13
E31 E21
3
2
2
2 4
3 5=4 0 1
7
0 1
3
2
1 5
5
3) ( 1)(equation 1)”
3 2
3
2
2 4
2
3 5=4 0 1
1 5
7
0 1
5
“Elimination matrices”
5
Elimination matrix : Second step
2x + 4y
2z = 2
y+z =4
elimination
2x + 4y
y+z =4
y + 5z = 12
2
1
4 0
?
eliminate
32
0 0
2
1 0 54 0
? ?
0
2z = 2
4z = 8
4
1
1
3
2
2
2 4
1 5=4 0 1
5
0 0
3
2
1 5
4
How do we express ”(equation 3) (1)(equation 2)”
Monday, October 7, 13
6
Elimination matrix : Second step
How do we express
2
32
1
0 0
4 0
1 0 54
0
1 1
`32
E32
”(equation 3) (1)(equation 2)”
3 2
3
2 4
2
2 4
2
0 1
1 5=4 0 1
1 5
0 1
5
0 0
4
E31 E21 A
U
We found a sequence of matrices that will take A to U
Monday, October 7, 13
7
Elimination matrices
We have constructed elimination matrices
2
1
4 0
0
Check
32
0 0
1 0 54
1 1
E32
E31 E21
2
32
1 0 0
2 1 0 54
1 0 1
E31 E21
32
1 0 0
= 4 0 1 0 54
1 0 1
2
4
2
3
4
9
3
2
2
2 4
3 5=4 0 1
7
0 0
U
A
3
2
1 0 0
2 1 0 5=4
0 0 1
3
2
1 5
4
3
1 0 0
2 1 0 5
1 0 1
E32 E31 E21 A = U
Product of elimination
matrices is lower triangular
Monday, October 7, 13
U is an upper triangular matrix
8