(0 )2 1436587@9BAC9EDGF4HI 7P5QHRFTSVU0DXW6F4H

46 CAPÍTULO 4. ¦ £§ ¥£ £¡ ¦§©££¢¡ £ §¥£
−−−−−→
ℓ3 − 2
3 ℓ2
⎡
⎢
⎣
r ([A8 | B8]) = 3
−5 −2 1 | −1
0 − 3
5
4
5 |
1
5
0 0
5
3 | − 4
3
0 0 0 | 0
0 0 0 | 0
⎤
⎥
⎦ = [A′ 8 | B ′ 8]
r (A8) = r ([A8 | B8]) = 3
(S8) ¡
¢ ¢ [A ′ 8 | B′ 8 ]
¢
⎡
[A ′ 8 | B ′ 8] =
−−−−−→
ℓ2 − 4
5 ℓ3
−−−−−→
ℓ1 − 2
3 ℓ2
⎢
⎣
⎡
⎢
⎣
⎡
⎢
⎣
−5 −2 1 | −1
0 − 3
5
4
5 |
1
5
0 0
5
3 | − 4
3
0 0 0 | 0
0 0 0 | 0
−5 −2 1 | −1
0 −3 0 |
21
5
0 0 5 | −4
0 0 0 | 0
0 0 0 | 0
−5 0 0 | −3
= [A8” | B8”]
0 −3 0 |
21
5
0 0 5 | −4
0 0 0 | 0
0 0 0 | 0
⎤
⎥
⎦
⎤
−−−−→
5ℓ2
3ℓ3
⎡
⎢
⎣
⎡
⎥
⎦ −−−−−→
ℓ1 − 1
5 ℓ3
⎢
⎣
⎤
⎥
⎦
−−−−−−−−−−→
− 1
5 ℓ1, − 1
3 ℓ2
1
5 ℓ3
A8”X = B8”
¥
⎡
5
⎢
⎣
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
⎤
⎥
⎦
x
y
z
⎡
⎢
= ⎢
⎣
3
− 7
5
− 4
5
0
0
⎤
⎥
⎦ ⇐⇒
x
−5 −2 1 | −1
0 −3 4 | 1
0 0 5 | −4
0 0 0 | 0
0 0 0 | 0
−5 −2 0 | − 1
5
0 −3 0 |
21
5
0 0 5 | −4
y
z
0 0 0 | 0
0 0 0 | 0
⎡
⎢
⎣
1 0 0 |
⎤
⎥
⎦
⎤
⎥
⎦
5
3
0 1 0 | − 7
5
0 0 5 | − 4
5
0 0 0 | 0
0 0 0 | 0
=
5
3
− 7
5
− 4
5
¡ (S9) £¢ ¢ ©
(S9)
⎧
⎪⎨
⎪⎩
x + y + z = −1
2x + y = 0
y + z = 2
x − z = −1
⇐⇒
2
3
1 1 1 2
6 2 1 0 7 4
4 0 1 1 5
1 0 −1
| {z }
A9
x
3
y 5
z
| {z }
X
⇐⇒ A9X = B9
=
⎤
⎥
⎦
2
6
4
−1
0
2
3
7
5
−1
| {z }
B9