Matemática discreta Ejemplos

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 1 & 1 & 1 & 260 \\ 25000 & 30000 & 40000 & 8000000 \end{array}]$

Paso 1

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

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Paso 1.1

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

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 1 - 1 & 1 + 2 & 1 - 0 & 260 - 0 \\ 25000 & 30000 & 40000 & 8000000 \end{array}]$

Paso 1.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 3 & 1 & 260 \\ 25000 & 30000 & 40000 & 8000000 \end{array}]$

Paso 2

Perform the row operation $R_{3} = R_{3} - 25000 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Paso 2.1

Perform the row operation $R_{3} = R_{3} - 25000 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 3 & 1 & 260 \\ 25000 - 25000 \cdot 1 & 30000 - 25000 \cdot - 2 & 40000 - 25000 \cdot 0 & 8000000 - 25000 \cdot 0 \end{array}]$

Paso 2.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 3 & 1 & 260 \\ 0 & 80000 & 40000 & 8000000 \end{array}]$

Paso 3

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

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Paso 3.1

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

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ \frac{0}{3} & \frac{3}{3} & \frac{1}{3} & \frac{260}{3} \\ 0 & 80000 & 40000 & 8000000 \end{array}]$

Paso 3.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 1 & \frac{1}{3} & \frac{260}{3} \\ 0 & 80000 & 40000 & 8000000 \end{array}]$

Paso 4

Perform the row operation $R_{3} = R_{3} - 80000 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Paso 4.1

Perform the row operation $R_{3} = R_{3} - 80000 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 1 & \frac{1}{3} & \frac{260}{3} \\ 0 - 80000 \cdot 0 & 80000 - 80000 \cdot 1 & 40000 - 80000 (\frac{1}{3}) & 8000000 - 80000 (\frac{260}{3}) \end{array}]$

Paso 4.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 1 & \frac{1}{3} & \frac{260}{3} \\ 0 & 0 & \frac{40000}{3} & \frac{3200000}{3} \end{array}]$

Paso 5

Multiply each element of $R_{3}$ by $\frac{3}{40000}$ to make the entry at $3, 3$ a $1$ .

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Paso 5.1

Multiply each element of $R_{3}$ by $\frac{3}{40000}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 1 & \frac{1}{3} & \frac{260}{3} \\ \frac{3}{40000} \cdot 0 & \frac{3}{40000} \cdot 0 & \frac{3}{40000} \cdot \frac{40000}{3} & \frac{3}{40000} \cdot \frac{3200000}{3} \end{array}]$

Paso 5.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 1 & \frac{1}{3} & \frac{260}{3} \\ 0 & 0 & 1 & 80 \end{array}]$

Paso 6

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

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Paso 6.1

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

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 - \frac{1}{3} \cdot 0 & 1 - \frac{1}{3} \cdot 0 & \frac{1}{3} - \frac{1}{3} \cdot 1 & \frac{260}{3} - \frac{1}{3} \cdot 80 \\ 0 & 0 & 1 & 80 \end{array}]$

Paso 6.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 \\ 0 & 1 & 0 & 60 \\ 0 & 0 & 1 & 80 \end{array}]$

Paso 7

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

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Paso 7.1

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

$[\begin{array}{c} 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 0 + 2 \cdot 0 & 0 + 2 \cdot 60 \\ 0 & 1 & 0 & 60 \\ 0 & 0 & 1 & 80 \end{array}]$

Paso 7.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 120 \\ 0 & 1 & 0 & 60 \\ 0 & 0 & 1 & 80 \end{array}]$