Álgebra linear Exemplos

$2 x + 4 y - 3 z = 1$ , $x + 2 y - 2 z = 0$ , $2 x + y + 3 z = 5$

Etapa 1

Escreva o sistema de equações em formato de matriz.

$[\begin{array}{c} 2 & 4 & - 3 & 1 \\ 1 & 2 & - 2 & 0 \\ 2 & 1 & 3 & 5 \end{array}]$

Etapa 2

Encontre a forma escalonada reduzida por linhas.

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Etapa 2.1

Multiply each element of $R_{1}$ by $\frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

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Etapa 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{2}{2} & \frac{4}{2} & \frac{- 3}{2} & \frac{1}{2} \\ 1 & 2 & - 2 & 0 \\ 2 & 1 & 3 & 5 \end{array}]$

Etapa 2.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 2 & - \frac{3}{2} & \frac{1}{2} \\ 1 & 2 & - 2 & 0 \\ 2 & 1 & 3 & 5 \end{array}]$

Etapa 2.2

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

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Etapa 2.2.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & - \frac{3}{2} & \frac{1}{2} \\ 1 - 1 & 2 - 2 & - 2 + \frac{3}{2} & 0 - \frac{1}{2} \\ 2 & 1 & 3 & 5 \end{array}]$

Etapa 2.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 2 & - \frac{3}{2} & \frac{1}{2} \\ 0 & 0 & - \frac{1}{2} & - \frac{1}{2} \\ 2 & 1 & 3 & 5 \end{array}]$

Etapa 2.3

Perform the row operation $R_{3} = R_{3} - 2 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Etapa 2.3.1

Perform the row operation $R_{3} = R_{3} - 2 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & - \frac{3}{2} & \frac{1}{2} \\ 0 & 0 & - \frac{1}{2} & - \frac{1}{2} \\ 2 - 2 \cdot 1 & 1 - 2 \cdot 2 & 3 - 2 (- \frac{3}{2}) & 5 - 2 (\frac{1}{2}) \end{array}]$

Etapa 2.3.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 2 & - \frac{3}{2} & \frac{1}{2} \\ 0 & 0 & - \frac{1}{2} & - \frac{1}{2} \\ 0 & - 3 & 6 & 4 \end{array}]$

Etapa 2.4

Swap $R_{3}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

$[\begin{array}{c} 1 & 2 & - \frac{3}{2} & \frac{1}{2} \\ 0 & - 3 & 6 & 4 \\ 0 & 0 & - \frac{1}{2} & - \frac{1}{2} \end{array}]$

Etapa 2.5

Multiply each element of $R_{2}$ by $- \frac{1}{3}$ to make the entry at $2, 2$ a $1$ .

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Etapa 2.5.1

Multiply each element of $R_{2}$ by $- \frac{1}{3}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 2 & - \frac{3}{2} & \frac{1}{2} \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot 6 & - \frac{1}{3} \cdot 4 \\ 0 & 0 & - \frac{1}{2} & - \frac{1}{2} \end{array}]$

Etapa 2.5.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 2 & - \frac{3}{2} & \frac{1}{2} \\ 0 & 1 & - 2 & - \frac{4}{3} \\ 0 & 0 & - \frac{1}{2} & - \frac{1}{2} \end{array}]$

Etapa 2.6

Multiply each element of $R_{3}$ by $- 2$ to make the entry at $3, 3$ a $1$ .

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Etapa 2.6.1

Multiply each element of $R_{3}$ by $- 2$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 2 & - \frac{3}{2} & \frac{1}{2} \\ 0 & 1 & - 2 & - \frac{4}{3} \\ - 2 \cdot 0 & - 2 \cdot 0 & - 2 (- \frac{1}{2}) & - 2 (- \frac{1}{2}) \end{array}]$

Etapa 2.6.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & 2 & - \frac{3}{2} & \frac{1}{2} \\ 0 & 1 & - 2 & - \frac{4}{3} \\ 0 & 0 & 1 & 1 \end{array}]$

Etapa 2.7

Perform the row operation $R_{2} = R_{2} + 2 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Etapa 2.7.1

Perform the row operation $R_{2} = R_{2} + 2 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 2 & - \frac{3}{2} & \frac{1}{2} \\ 0 + 2 \cdot 0 & 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - \frac{4}{3} + 2 \cdot 1 \\ 0 & 0 & 1 & 1 \end{array}]$

Etapa 2.7.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & 2 & - \frac{3}{2} & \frac{1}{2} \\ 0 & 1 & 0 & \frac{2}{3} \\ 0 & 0 & 1 & 1 \end{array}]$

Etapa 2.8

Perform the row operation $R_{1} = R_{1} + \frac{3}{2} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Etapa 2.8.1

Perform the row operation $R_{1} = R_{1} + \frac{3}{2} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 + \frac{3}{2} \cdot 0 & 2 + \frac{3}{2} \cdot 0 & - \frac{3}{2} + \frac{3}{2} \cdot 1 & \frac{1}{2} + \frac{3}{2} \cdot 1 \\ 0 & 1 & 0 & \frac{2}{3} \\ 0 & 0 & 1 & 1 \end{array}]$

Etapa 2.8.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 2 & 0 & 2 \\ 0 & 1 & 0 & \frac{2}{3} \\ 0 & 0 & 1 & 1 \end{array}]$

Etapa 2.9

Perform the row operation $R_{1} = R_{1} - 2 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Etapa 2.9.1

Perform the row operation $R_{1} = R_{1} - 2 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & 2 - 2 (\frac{2}{3}) \\ 0 & 1 & 0 & \frac{2}{3} \\ 0 & 0 & 1 & 1 \end{array}]$

Etapa 2.9.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & \frac{2}{3} \\ 0 & 1 & 0 & \frac{2}{3} \\ 0 & 0 & 1 & 1 \end{array}]$

Etapa 3

Use a matriz de resultados para declarar as soluções finais ao sistema de equações.

$x = \frac{2}{3}$

$y = \frac{2}{3}$

$z = 1$

Etapa 4

A solução é o conjunto de pares ordenados que tornam o sistema verdadeiro.

$(\frac{2}{3}, \frac{2}{3}, 1)$

Etapa 5

Para decompor um vetor da solução, reorganize cada equação representada na forma de linha reduzida da matriz aumentada resolvendo a variável dependente em cada linha que produz igualdade vetorial.

$X = [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} \frac{2}{3} \\ \frac{2}{3} \\ 1 \end{array}]$