Soluções dos Exercícios de AN Lista I - Sistemas Lineares

Ax = b m n
???
• m < n
• m = n
• m > n
b
A
⎧
⎪⎨
⎪⎩
x1 + 6x2 + 2x3 + 4x4 = 8
3x1 + 19x2 + 4x3 + 15x4 = 25
1x1 + 4x2 + 8x3 − 12x4 = 18
5x1 + 33x2 + 9x3 + 3x4 = 72
Ax = b
Ux = c

Ax = b
LUx = b
Ax = b
⎛
⎜
⎝
1
3
1
6
19
4
2
4
8
⎞ ⎛ ⎞
4 x1
15 ⎟ ⎜ ⎟
⎟ ⎜ x2 ⎟
⎟ ⎜ ⎟
−12 ⎠ ⎝ x3 ⎠
5 33 9 3 x4
=
⎛
⎜
⎝
⎛
⎜
⎝
1 6 2 4 8
3 19 4 15 25
1 4 8 −12 18
5 33 9 3 72
⎞
⎟
⎠ .
x4 = 1,
⎛
⎜
⎝
1 6 2 4 8
0 1 −2 3 1
0 0 2 −10 12
0 0 0 −1 −1
⎞
⎟
⎠
⎛
1
⎜ 0
⎜
⎝ 0
6
1
0
2
−2
2
⎞ ⎛ ⎞
4 x1
3 ⎟ ⎜ ⎟
⎟ ⎜ x2 ⎟
⎟ ⎜ ⎟
−10 ⎠ ⎝ x3 ⎠
0 0 0 −1 x4
=
⎛
⎜
⎝
8
25
18
72
8
1
12
−1
⎞
⎟
⎠
⎞
⎟
⎠ .
x3 = 6 + 5x4 = 6 + 5 × 1 = 11,
x2 = 1 − 3x4 + 2x3 = 1 − 3 × 1 + 2 × 11 = 20,
x1 = 8 − 4x4 − 2x3 − 6x2 = 8 − 4 × 1 − 2 × 11 − 6 × 20 = −138.

⎛
⎜
L = ⎜
⎝
1 0 0 0
3 1 0 0
1 −2 1 0
5 3 5
2 1
⎞
⎟
⎠ .
LUx = b
⎛
⎜
⎝
1 0 0 0
3 1 0 0
1 −2 1 0
5 3 5
2 1
⎞ ⎛
1
⎟ ⎜
⎟ ⎜ 0
⎟ ⎜
⎠ ⎝ 0
6
1
0
2
−2
2
⎞ ⎛ ⎞
4 x1
3 ⎟ ⎜ ⎟
⎟ ⎜ x2 ⎟
⎟ ⎜ ⎟
−10 ⎠ ⎝ x3 ⎠
0 0 0 −1 x4
=
⎛
⎜
⎝
⎧
⎪⎨
⎪⎩
x1 − 3x2 + 2x3 = 11
−2x1 + 8x2 − 1x3 = −15
4x1 − 6x2 + 5x3 = 29
⎛
⎜
⎝
1 −3 2
−2 8 −1
4 −6 5
⎞ ⎛
⎟ ⎜
⎠ ⎝
x1
x2
⎞ ⎛
⎟ ⎜
⎠ = ⎝
⎞
11
⎟
−15 ⎠
29
det(A) = −4 = 0 LU = P A
A
M1 =
P1 =
⎛
⎜
⎝
⎛
⎜
⎝
0 0 1
0 1 0
1 0 0
1 0 0
1/2 1 0
−1/4 0 1
⎞
x3
⎛
8
25
18
72
⎞
⎟
⎠ .
4 −6 5
−2 8 −1
⎟
⎠ ⇒
⎜
P1A ⎝
⎟
⎠
⎞
⎟
⎠ ⇒
1 −3 2
⎛
4 −6
⎜
M1P1A ⎝ 0 5
⎞
5
⎟
3/2 ⎠
0 −3/2 3/4
⎞

L = (M2M1) −1 ⎜
⎝
M2 =
⎛
⎛
⎜
⎝
1 0 0
0 1 0
0 3/10 1
1 0 0
−1/2 1 0
1/4 −3/10 1
⎞
⎛
⎟
⎜
⎠ ⇒ M2M1P1A ⎝
P = P1
Lc = P b
⎛
⎜
⎝
1 0 0
−1/2 1 0
1/4 −3/10 1
⎞
⎟
⎠
⎞ ⎛
⎟ ⎜
⎠ ⎝
c1
c2
⎞ ⎛
⎟ ⎜
⎠ = ⎝
⎞
29
⎟
−15 ⎠
11
c3
c1 = 29,
c2 = −1/2,
c3 = 18/5.
Ux = c
⎛
⎜
⎝
4 −6 5
0 5 3/2
0 0 6/5
⎞ ⎛
⎟ ⎜
⎠ ⎝
x1
x2
⎞ ⎛
⎟ ⎜
⎠ = ⎝
⎞
29
⎟
−1/2 ⎠
18/5
x3
x3 = 3,
x2 = −1,
x1 = 2
4 −6 5
0 5 3/2
0 0 6/5
⎞
⎟
⎠ = U
⎧
⎪⎨
⎪⎩
9x1 + 6x2 − 3x3 + 3x4 = 12
6x1 + 20x2 + 2x3 + 22x4 = 64
−3x1 + 2x2 + 6x3 + 2x4 = 4
3x1 + 22x2 + 2x3 + 28x4 = 82
A = LL t
Ly = b y L t x = y

A = LL t A
⎛
⎜
⎝
9 6 −3 3
6 20 2 22
−3 2 6 2
3 22 2 28
⎞
⎛
⎟
⎠ ⇒
⎜
⎝
⎛
⎜
⇒ ⎜
⎝
3 0 0 0
2 20 0 0
−1 2 6 0
1 22 2 28
3 0 0 0
2 4 0 0
−1 1 4 0
1 5 −2 2
⎛
L t ⎜
= ⎜
⎝
⎞
⎞
⎛
⎟
⎠ ⇒
⎜
⎝
⎛
⎟
⎠ ⇒
⎜
⎝
3 2 −1 1
0 4 1 5
0 0 2 −1
0 0 0 1
⎞
⎟
⎠
3 0 0 0
2 16 0 0
−1 4 5 0
1 20 3 27
3 0 0 0
2 4 0 0
−1 1 2 0
1 5 −1 1
Ly = b
⎛
⎜
⎝
3
2
−1
0
4
1
0
0
2
⎞ ⎛ ⎞
0 y1
0 ⎟ ⎜ ⎟
⎟ ⎜ y2 ⎟
⎟ ⎜ ⎟
0 ⎠ ⎝ y3 ⎠
1 5 −1 1 y4
=
⎛
⎜
⎝
y1 = 4,
y2 = 14,
y3 = −3
y4 = 5.
12
64
4
82
L t x = y
⎛
⎜
⎝
3
0
0
2
4
0
−1
1
2
⎞ ⎛ ⎞
1 x1
5 ⎟ ⎜ ⎟
⎟ ⎜ x2 ⎟
⎟ ⎜ ⎟
−1 ⎠ ⎝ x3 ⎠
0 0 0 1 x4
=
⎛
⎜
⎝
4
14
−3
5
⎞
⎟
⎠
⎞
⎟
⎠
⎞
⎞
⎟
⎠
⎟ = L
⎠

x4 = 5
x3 = 1,
x2 = −3,
x1 = 2

x4 = 5
x3 = 1,
x2 = −3,
x1 = 2

x4 = 5
x3 = 1,
x2 = −3,
x1 = 2