Question

(Álgebra Linear) Encontrar a solução do Sistema Linear (se existir) através de matriz linha reduzida na forma Escada (indicar as operações elementares usadas)

x + y + z + w = 10
x - y - z + w = 0
2x + y + z + w = -1
2x + 2y - 3z - 3w = -15

Answer 1

$\begin{pmatrix}x&y&z&w&10\\ x&-y&-z&w&0\\ 2x&y&z&w&-1\\ 2x&2y&-3z&-3w&-15\end{pmatrix}$

*vamos trocar a linha 1 pela 4

$\begin{pmatrix}2x&2y&-3z&-3w&-15\\ x&-y&-z&w&0\\ 2x&y&z&w&-1\\ x&y&z&w&10\end{pmatrix}$

*Agora zerar a primeira colula

$\:L_2-\frac{1}{2}\cdot \:L_1\\\:L_3-1 .\:L_1\\\:L_4-\frac{1}{2}\cdot \:L_1$

$\begin{pmatrix}2x&2y&-3z&-3w&-15\\ 0&-2y&\frac{z}{2}&\frac{5w}{2}&\frac{15}{2}\\ 0&-y&4z&4w&14\\ 0&0&\frac{5z}{2}&\frac{5w}{2}&\frac{35}{2}\end{pmatrix}$

$\begin{pmatrix}2x&2y&-3z&-3w&-15\\ 0&-y&4z&4w&14\\ 0&-2y&\frac{z}{2}&\frac{5w}{2}&\frac{15}{2}\\ 0&0&\frac{5z}{2}&\frac{5w}{2}&\frac{35}{2}\end{pmatrix}$

$\:L_3-2\cdot \:L_2$

$\begin{pmatrix}2x&2y&-3z&-3w&-15\\ 0&-y&4z&4w&14\\ 0&0&-\frac{15z}{2}&-\frac{11w}{2}&-\frac{41}{2}\\ 0&0&\frac{5z}{2}&\frac{5w}{2}&\frac{35}{2}\end{pmatrix}$

$\begin{pmatrix}2x&2y&-3z&-3w&-15\\ 0&-y&4z&4w&14\\ 0&0&\frac{5z}{2}&\frac{5w}{2}&\frac{35}{2}\\ 0&0&-\frac{15z}{2}&-\frac{11w}{2}&-\frac{41}{2}\end{pmatrix}$

$\:L_4-3\cdot \:L_3$

$\begin{pmatrix}2x&2y&-3z&-3w&-15\\ 0&-y&4z&4w&14\\ 0&0&\frac{5z}{2}&\frac{5w}{2}&\frac{35}{2}\\ 0&0&0&2w&32\end{pmatrix}$

2w=32

w=16

5z+5(16)=35

5z=35-80

z=-9

-y+4(-9)+4(16)=14

-y-36+64=14

-y=14+36-64

y=14

2x+2(14)-3(-9)-3(16)=-15

2x+28+27-48=-15

2x=-15-7

x=-11