Álgebra linear Exemplos

$[\begin{array}{c} a & b & c & 1000 \\ 100 a & 200 b & 500 c & 300000 \end{array}]$

Etapa 1

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

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Etapa 1.1

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

$[\begin{array}{c} \frac{a}{a} & \frac{b}{a} & \frac{c}{a} & \frac{1000}{a} \\ 100 a & 200 b & 500 c & 300000 \end{array}]$

Etapa 1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{c}{a} & \frac{1000}{a} \\ 100 a & 200 b & 500 c & 300000 \end{array}]$

Etapa 2

Perform the row operation $R_{2} = R_{2} - 100 a R_{1}$ to make the entry at $2, 1$ a $0$ .

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Etapa 2.1

Perform the row operation $R_{2} = R_{2} - 100 a R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{c}{a} & \frac{1000}{a} \\ 100 a - 100 a \cdot 1 & 200 b - 100 a \frac{b}{a} & 500 c - 100 a \frac{c}{a} & 300000 - 100 a \frac{1000}{a} \end{array}]$

Etapa 2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{c}{a} & \frac{1000}{a} \\ 0 & 100 b & 400 c & 200000 \end{array}]$

Etapa 3

Multiply each element of $R_{2}$ by $\frac{1}{100 b}$ to make the entry at $2, 2$ a $1$ .

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Etapa 3.1

Multiply each element of $R_{2}$ by $\frac{1}{100 b}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{c}{a} & \frac{1000}{a} \\ \frac{0}{100 b} & \frac{100 b}{100 b} & \frac{400 c}{100 b} & \frac{200000}{100 b} \end{array}]$

Etapa 3.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{c}{a} & \frac{1000}{a} \\ 0 & 1 & \frac{4 c}{b} & \frac{2000}{b} \end{array}]$

Etapa 4

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

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Etapa 4.1

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

$[\begin{array}{c} 1 - \frac{b}{a} \cdot 0 & \frac{b}{a} - \frac{b}{a} \cdot 1 & \frac{c}{a} - \frac{b}{a} \cdot \frac{4 c}{b} & \frac{1000}{a} - \frac{b}{a} \cdot \frac{2000}{b} \\ 0 & 1 & \frac{4 c}{b} & \frac{2000}{b} \end{array}]$

Etapa 4.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - \frac{3 c}{a} & - \frac{1000}{a} \\ 0 & 1 & \frac{4 c}{b} & \frac{2000}{b} \end{array}]$