Álgebra linear Exemplos

$x e^{2 t} [\begin{array}{c} 1 \\ 0 \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1

Simplifique.

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Etapa 1.1

Simplifique cada termo.

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Etapa 1.1.1

Multiplique $x e^{2 t}$ por cada elemento da matriz.

$[\begin{array}{c} x e^{2 t} \cdot 1 \\ x e^{2 t} \cdot 0 \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1.1.2

Simplifique cada elemento da matriz.

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Etapa 1.1.2.1

Multiplique $x$ por $1$ .

$[\begin{array}{c} x e^{2 t} \\ x e^{2 t} \cdot 0 \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1.1.2.2

Multiplique $x e^{2 t} \cdot 0$ .

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Etapa 1.1.2.2.1

Multiplique $0$ por $x$ .

$[\begin{array}{c} x e^{2 t} \\ 0 e^{2 t} \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1.1.2.2.2

Multiplique $0$ por $e^{2 t}$ .

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1.1.3

Multiplique $y e^{2 t}$ por cada elemento da matriz.

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} y e^{2 t} \cdot 0 \\ y e^{2 t} \cdot - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1.1.4

Simplifique cada elemento da matriz.

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Etapa 1.1.4.1

Multiplique $y e^{2 t} \cdot 0$ .

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Etapa 1.1.4.1.1

Multiplique $0$ por $y$ .

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 e^{2 t} \\ y e^{2 t} \cdot - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1.1.4.1.2

Multiplique $0$ por $e^{2 t}$ .

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ y e^{2 t} \cdot - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1.1.4.2

Mova $- 1$ para a esquerda de $y e^{2 t}$ .

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - 1 \cdot (y e^{2 t}) \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1.1.4.3

Reescreva $- 1 (y e^{2 t})$ como $- (y e^{2 t})$ .

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1.1.5

Multiplique $t$ por cada elemento da matriz.

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \cdot 1 \\ t \cdot 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1.1.6

Simplifique cada elemento da matriz.

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Etapa 1.1.6.1

Multiplique $t$ por $1$ .

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ t \cdot 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1.1.6.2

Multiplique $t$ por $0$ .

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1.2

Adicione os elementos correspondentes.

$[\begin{array}{c} x e^{2 t} + 0 \\ 0 - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1.3

Simplify each element.

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Etapa 1.3.1

Some $x e^{2 t}$ e $0$ .

$[\begin{array}{c} x e^{2 t} \\ 0 - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1.3.2

Subtraia $y e^{2 t}$ de $0$ .

$[\begin{array}{c} x e^{2 t} \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1.4

Adicione os elementos correspondentes.

$[\begin{array}{c} x e^{2 t} + t \\ - y e^{2 t} + 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 1.5

Some $- y e^{2 t}$ e $0$ .

$[\begin{array}{c} x e^{2 t} + t \\ - y e^{2 t} \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

Etapa 2

A equação da matriz pode ser escrita como um conjunto de equações.

$x e^{2 t} + t = 0$

$- y e^{2 t} = 1$