APS2 algebra

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FACULDADE DE AMERICANA 
SISTEMAS DE INFORMAÇÃO/ CIÊNCIAS DA COMPUTAÇÃO 
 
 
Dayane Corrêa da Silva RA: 20202387 
Leonardo Menossi RA: 20201394 
 
 
 
 
 
 
 
APS 2 – Lista de Exercícios 1 
 
 
 
 
 
 
 
 
 
 
 
AMERICANA-SP 
2021 
1) 
 
 
 
 
Eixo x: 𝐷1 + 𝐷2𝑥 = 𝐷1 + 𝐷2 ∙ 𝑐𝑜𝑠𝜃 = (200) + (400) ∙ (−0,71) → 𝐷𝑥 = 83𝑘𝑚 
Eixo y: 𝐷𝑦 = 𝐷3 + 𝐷2𝑥 = 𝐷3 + 𝐷2 ∙ 𝑐𝑜𝑠𝜃 = (500) + (400) ∙ (−0,71) → 𝐷𝑦 =
783𝑘𝑚 
Deslocamento: 
𝐷 = (83𝑖 − 783𝑗)𝑘𝑚 
𝑡𝑎𝑔𝛽 =
𝐷𝑥
𝐷𝑦
=
783
83
= −9,44 
𝛽 = 𝑎𝑟𝑐𝑡𝑎𝑔(9,44) → 𝛽 = 83,95° 
 
𝐷 = √(83)2 + (783)2 
𝐷 = 787 𝑘𝑚 
 
2) 𝐴 = (𝑎𝑖𝑗)3×3 𝑎𝑖𝑗 = 𝑖
2 − 4 ∙ 𝑗 
[
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
] = [
−3 −7 −11
0 −4 −8
5 1 −4
] 
 
𝑎11 = (1)
2 − 4.1 = −3 
𝑎21 = (2)
2 − 4.1 = 0 
𝑎31 = (3)
2 − 4.1 = 5 
 
𝑎12 = (1)
2 − 4.2 = −7 
𝑎22 = (2)
2 − 4.2 = −4 
𝑎32 = (3)
2 − 4.2 = 1 
 
𝑎13 = (1)
2 − 4.3 = −11 
𝑎23 = (2)
2 − 4.3 = −8 
𝑎33 = (3)
2 − 4.3 = −4 
 
y 
x 
D2x 
D2y 
D2 
3) [
1 0 1 − 2𝑥
2𝑦 + 2 1 0
0 0 𝑥 +
3
2
] 
𝑥13 → 1 − 2𝑥 → 2𝑥 = 1 → 𝑥 =
1
2
 
𝑥21 → 2𝑦 + 2 → 2𝑦 = −2 → 𝑦 = −
2
2
→ 𝑡 = −1 
𝑥31 → 𝑥 +
3
2
→ 𝑥 = 1 −
3
2
→ 𝑥 =
2 − 3
2
→ 𝑥 = −
1
2
 
 
 
4) 𝐸 = [
−2 + 2 3 + 2
1 + (−3) 4 + (−1)
2 + 0 −3 + (−1)
] 
𝐸 = [
0 5
−2 3
2 −4
] 
 
𝐹 = [
0 − 2 1 − (−3) −1 − 0
2 − (−1) 3 − 2 −2 − 1
] 
𝐹 = [
−2 4 −1
3 1 −3
] 
 
5) a) 𝐸 = [
2 2
−3 −1
0 −1
] × [
0 1 −1
2 3 −2
] 
= [
2 × 0 + 2 × 2 2 × 1 + 2 × 3 2 × (−1) + 2 × (−2)
−3 × 0 + (−1) × 2 −3 × 1 + (−1) × 3 −3 × (−1) + (−1) × (−2)
0 × 0 + (−1) × 2 0 × 1 + (−1) × 3 0 × (−1) + (−1) × (−2)
] 
= [
4 8 −6
−2 −6 5
−2 −3 2
] 
 
b) 𝐹 = [
2 −3 0
−1 2 1
] × [
−2 3
1 4
2 −3
] 
= [
2 × (−2) + (−3) × 1 + 0 × 2 2 × 3 + (−3) × 4 + 0 × (−3)
−1 × (−2) + 2 × 1 + 1 × 2 −1 × 3 + 2 × 4 + 1 × (−3)
] 
= [
−7 −6
6 2
] 
 
6) [
1 −1
2 3
] = [
1 −1
2 3
|
1 0
0 1
] 
= [
1 −1
1 × (−2) + 2 −1 × (−2) + 3
|
1 0
1 × (−2) + 0 0 × (−2) + 1
] 
= [
1 −1
0 5
|
1 0
−2 1
] 
= [
1 −1
0
5
5
5
|
1 0
−
2
5
1
5
] 
= [0 + 1 1 +
(−1)
0 1
|
−
2
5
+ 1
1
5
+ 0
−
2
5
1
5
] 
= [
1 0
0 1
|
3
5
1
5
−
2
5
1
5
] 
= [
3
5
1
5
−
2
5
1
5
] 
 
7) 𝑋 = [
1 1
1 1
] = 1 ∙ 1 − 1 ∙ 1 = 0 
 
𝑌 = [
0 −2 −3
2 3 1
−1 −2 2
] 
𝑌 = 0 ∙ |
3 1
−2 2
| − 2 ∙ |
−2 −3
−2 2
| + (−1) ∙ |
−2 −3
3 1
| 
𝑌 = 0 − 2 ∙ (−10) + (−1) ∙ 7 
𝑌 = −20 − 7 
𝑌 = 13 
 
𝑍 = [
0
−1
−2
 
2
3
1
 
1
1
2
 
1
−3
3
] 
 
 
8) {
𝑥 + 3𝑦 − 2𝑧 = 2
−2𝑥 + 𝑦 − 𝑧 = 1
3𝑥 + 2𝑦 + 𝑧 = 3
 
𝐷 = |
1 3 −2
−2 1 −1
3 2 1
|
1 3
−2 1
3 2
 = 14 
𝐷1 = |
2 3 −2
1 1 −1
3 2 1
|
2 3
1 1
3 2
= −4 
𝐷2 = |
1 2 −2
−2 1 −1
3 3 1
|
1 2
−2 1
3 3
= 20 
𝐷3 = |
1 3 2
−2 1 1
3 2 3
|
1 3
−2 1
3 2
= 14 
 
𝑥 = −
4
14
= −
2
7
 
𝑦 =
20
14
=
10
7
 
𝑧 =
14
14
= 1 
 
 
9) {
2𝑥 + 𝑦 + 𝑧 = 2
−𝑥 − 𝑦 − 𝑧 = −1
−3𝑥 − 2𝑦 − 2𝑧 = 2
 
{
2𝑥 + 𝑦 + 𝑧 = 2
𝑥 = 1 − 𝑦 − 𝑧
−3𝑥 − 2𝑦 − 2𝑧 = 2
 
{
2(1 − 𝑦 − 𝑧) + 𝑦 + 𝑧 = 2
−3(1 − 𝑦 − 𝑧) − 2𝑦 − 2𝑧 = 2
 
{
−𝑦 − 𝑧 = 0
𝑦 + 𝑧 = 5
 
0 = 5 
(𝑥, 𝑦, 𝑧) ∈ ∅