EJERCICIOS METODO GAUSS JORDAN

1. RAMON SHETT ARROYO
SEGUNDO BGU “B”
MATEMATICAS/ SISTEMA DE ECUACIONES POR METODO GAUSS-JORDAN
1
𝟐𝒙 + 𝒚 + 𝟐𝒛 = 𝟔
𝟔𝒙 − 𝟔𝒚 + 𝟔𝒛 = 𝟑𝟔
𝟒𝒙 + 𝟑𝒚 + 𝟑𝒛 = −𝟏
|
2 1 2
6 −6 6
4 3 3
|
6
36
−1
|
𝐹2
6
𝐹2 ↔ 𝐹1
→ |
1 −1 1
2 1 2
4 3 3
|
6
6
−1
|
𝐹2 − 2𝐹1
→ |
1 −1 1
0 3 0
4 3 3
|
6
−6
−1
|
𝐹3 − 4𝐹1
→
|
1 −1 1
0 3 0
0 7 −1
|
6
−6
−25
|
3𝐹3 − 7𝐹2
→ |
1 −1 1
0 3 0
0 0 −3
|
6
−6
−33
|
𝐹2
3
−
𝐹3
3
→ |
1 −1 1
0 1 0
0 0 1
|
6
−2
11
|
𝐹1 + 𝐹2
→
|
1 0 1
0 1 0
0 0 1
|
4
−2
11
|
𝐹1 − 𝐹3
→ |
1 0 0
0 1 0
0 0 1
|
−7
−2
11
|
𝑹𝒆𝒔𝒑𝒖𝒆𝒔𝒕𝒂: 𝒙 = −𝟕 𝒚 = −𝟐 𝒛 = 𝟏𝟏
𝟒𝒙 − 𝟐𝒚 + 𝟔𝒛 = 𝟖
𝒙 + 𝒚 − 𝟑𝒛 = −𝟏
𝟏𝟓𝒙 − 𝟑𝒚 + 𝟗𝒛 = 𝟐𝟏
|
4 −2 6
1 1 −3
15 −3 9
|
8
−1
21
|
𝐹2 ↔ 𝐹1
→ |
1 1 −3
4 −2 6
15 −3 9
|
−1
8
21
|
𝐹2 − 4𝐹1
→ |
1 1 −3
0 −6 18
15 −3 9
|
−1
13
21
|
𝐹3 − 15𝐹1
→ |
1 1 −3
0 −6 18
0 −18 54
|
−1
13
35
|
𝐹3 − 3𝐹2
→ |
1 1 −3
0 −6 18
0 0 0
|
−1
13
−4
|
𝑹𝒆𝒔𝒑𝒖𝒆𝒔𝒕𝒂:𝑵𝒐 𝒕𝒊𝒆𝒏𝒆 𝒔𝒐𝒍𝒖𝒄𝒊ó𝒏

2. RAMON SHETT ARROYO
SEGUNDO BGU “B”
MATEMATICAS/ SISTEMA DE ECUACIONES POR METODO GAUSS-JORDAN
2
𝒙 + 𝟐𝒚 + 𝟑𝒛 = 𝟏
𝟐𝒙 + 𝟑𝒚 + 𝟐𝒛 = 𝟐
𝟑𝒙 + 𝟑𝒚 + 𝟒𝒛 = 𝟏
|
1 2 3
2 3 2
3 3 4
|
1
2
1
|
𝐹2 − 2𝐹1
𝐹3 − 3𝐹1
→ |
1 2 3
0 −1 −4
0 −3 −5
|
1
0
−2
|
𝐹3 − 3𝐹2
→ |
1 2 3
0 −1 −4
0 0 7
|
1
0
−2
|
7𝐹2 + 4𝐹3
→ |
1 2 3
0 −7 0
0 0 7
|
1
−8
−2
|
−
𝐹2
7
𝐹3
7
→ |
1 2 3
0 1 0
0 0 1
|
1
8/7
−2/7
|
𝐹1 − 2𝐹2
→
|
1 0 3
0 1 0
0 0 1
|
−9/7
8/7
−2/7
|
𝐹1 − 3𝐹3
→ |
1 0 0
0 1 0
0 0 1
|
−3/7
8/7
−2/7
|
𝑹𝒆𝒔𝒑𝒖𝒆𝒔𝒕𝒂: 𝒙 = −
𝟑
𝟕
𝒚 =
𝟖
𝟕
𝒛 = −
𝟐
𝟕
𝒙 + 𝒚 + 𝒛 = 𝟔
𝒙 − 𝒚 + 𝒛 = 𝟐
𝟐𝒙 + 𝒚 − 𝒛 = 𝟏
|
1 1 1
1 −1 1
2 1 −1
|
6
2
1
|
𝐹2 − 𝐹1
𝐹3 − 2𝐹1
→ |
1 1 1
0 −2 0
0 −1 −3
|
6
−4
−11
|
2𝐹3 − 𝐹2
→ |
1 1 1
0 −2 0
0 0 −6
|
6
−4
−18
|
−
𝐹2
2
−
𝐹3
6
→ |
1 1 1
0 1 0
0 0 1
|
6
2
3
|
𝐹1 − 𝐹2
→ |
1 0 1
0 1 0
0 0 1
|
𝐹1 − 𝐹3
→ |
1 0 0
0 1 0
0 0 1
|
1
2
3
|
𝑹𝒆𝒔𝒑𝒖𝒆𝒔𝒕𝒂: 𝒙 = 𝟏 𝒚 = 𝟐 𝒛 = 𝟑

3. RAMON SHETT ARROYO
SEGUNDO BGU “B”
MATEMATICAS/ SISTEMA DE ECUACIONES POR METODO GAUSS-JORDAN
3
𝟕𝒙 − 𝟒𝒚 = 𝟏𝟐 → 7𝑥 − 4𝑦 + 0𝑧 = 12
−𝟒𝒙 + 𝟏𝟐𝒚− 𝟔𝒛 = 𝟎 → −4𝑥 + 12𝑦 − 6𝑧 = 0
−𝟔𝒚 + 𝟏𝟒𝒛 = 𝟎 → 0𝑥 + −6𝑦 + 14𝑧 = 0
|
7 −4 0
−4 12 −6
0 −6 14
|
12
0
0
|
𝐹1 + 2𝐹2
→ |
−1 20 −12
−4 12 −6
0 −6 14
|
12
0
0
|
−𝐹1
→ |
1 −20 12
−4 12 −6
0 −6 14
|
−12
0
0
|
𝐹2 + 4𝐹1
→ |
1 −20 12
0 −68 42
0 −6 14
|
−12
−48
0
|
68𝐹3 − 6𝐹2
→ |
1 −20 12
0 −68 42
0 0 700
|
−12
−48
288
|
𝐹3
700 →
|
1 −20 12
0 −68 42
0 0 1
|
−12
−48
72/175
|
𝐹2 − 42𝐹3
→ |
1 −20 12
0 −68 0
0 0 1
| −
−12
11424/175
72/175
|
−
𝐹2
68 →
|
1 −20 12
0 1 0
0 0 1
|
−12
168/175
72/175
|
𝐹1 + 20𝐹2
→ |
1 0 12
0 1 0
0 0 1
|
252/35
168/175
72/175
|
𝐹1 − 12𝐹3
→
|
1 0 0
0 1 0
0 0 1
|
396/175
168/175
72/175
|
𝑹𝒆𝒔𝒑𝒖𝒆𝒔𝒕𝒂: 𝒙 =
𝟑𝟗𝟔
𝟏𝟕𝟓
𝒚 =
𝟏𝟔𝟖
𝟏𝟕𝟓
𝒛 =
𝟕𝟐
𝟏𝟕𝟓
𝒙 + 𝟐𝒚 + 𝟑𝒛 = 𝟗
𝟒𝒙 + 𝒚 + 𝟐𝒛 = 𝟔
−𝟐𝒙 + 𝟓𝒚+ 𝟒𝒛 = −𝟐
|
1 2 3
4 1 2
−2 5 4
|
9
6
−2
|
𝐹2 − 4𝐹1
𝐹3 + 2𝐹1
→ |
1 2 3
0 −7 −10
0 9 10
|
9
−30
16
|
7𝐹3 + 9𝐹2
→ |
1 2 3
0 −7 −10
0 0 −20
|
9
−30
−158
|

4. RAMON SHETT ARROYO
SEGUNDO BGU “B”
MATEMATICAS/ SISTEMA DE ECUACIONES POR METODO GAUSS-JORDAN
4
−
𝐹3
20 → |
1 2 3
0 −7 −10
0 0 1
|
9
−30
79/10
|
𝐹2 + 10𝐹3
→ |
1 2 3
0 −7 0
0 0 1
|
9
49
79/10
|
−
𝐹2
7 →
|
1 2 3
0 1 0
0 0 1
|
9
−7
79/10
|
𝐹1 − 2𝐹2
→ |
1 0 3
0 1 0
0 0 1
|
23
−7
79/10
|
𝐹1 − 2𝐹3
→ |
1 0 0
0 1 0
0 0 1
|
−7/10
−7
79/10
|
𝑹𝒆𝒔𝒑𝒖𝒆𝒔𝒕𝒂: 𝒙 = −
𝟕
𝟏𝟎
𝒚 = −𝟕 𝒛 =
𝟕𝟗
𝟏𝟎