Álgebra lineal Ejemplos

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 1 & 2 & - 3 & 0 & - 5 & 0 \\ 1 & 3 & - 4 & 0 & - 13 & - 11 \\ 1 & - 1 & 1 & 1 & - 10 & 0 \\ 1 & 4 & - 5 & 0 & - 1 & 23 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 1 - 1 & 2 - 1 & - 3 + 2 & 0 - 0 & - 5 + 3 & 0 + 1 \\ 1 & 3 & - 4 & 0 & - 13 & - 11 \\ 1 & - 1 & 1 & 1 & - 10 & 0 \\ 1 & 4 & - 5 & 0 & - 1 & 23 \end{array}]$

Paso 1.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 1 & 3 & - 4 & 0 & - 13 & - 11 \\ 1 & - 1 & 1 & 1 & - 10 & 0 \\ 1 & 4 & - 5 & 0 & - 1 & 23 \end{array}]$

Paso 2

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

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Paso 2.1

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

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 1 - 1 & 3 - 1 & - 4 + 2 & 0 - 0 & - 13 + 3 & - 11 + 1 \\ 1 & - 1 & 1 & 1 & - 10 & 0 \\ 1 & 4 & - 5 & 0 & - 1 & 23 \end{array}]$

Paso 2.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 2 & - 2 & 0 & - 10 & - 10 \\ 1 & - 1 & 1 & 1 & - 10 & 0 \\ 1 & 4 & - 5 & 0 & - 1 & 23 \end{array}]$

Paso 3

Perform the row operation $R_{4} = R_{4} - R_{1}$ to make the entry at $4, 1$ a $0$ .

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Paso 3.1

Perform the row operation $R_{4} = R_{4} - R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 2 & - 2 & 0 & - 10 & - 10 \\ 1 - 1 & - 1 - 1 & 1 + 2 & 1 - 0 & - 10 + 3 & 0 + 1 \\ 1 & 4 & - 5 & 0 & - 1 & 23 \end{array}]$

Paso 3.2

Simplifica $R_{4}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 2 & - 2 & 0 & - 10 & - 10 \\ 0 & - 2 & 3 & 1 & - 7 & 1 \\ 1 & 4 & - 5 & 0 & - 1 & 23 \end{array}]$

Paso 4

Perform the row operation $R_{5} = R_{5} - R_{1}$ to make the entry at $5, 1$ a $0$ .

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Paso 4.1

Perform the row operation $R_{5} = R_{5} - R_{1}$ to make the entry at $5, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 2 & - 2 & 0 & - 10 & - 10 \\ 0 & - 2 & 3 & 1 & - 7 & 1 \\ 1 - 1 & 4 - 1 & - 5 + 2 & 0 - 0 & - 1 + 3 & 23 + 1 \end{array}]$

Paso 4.2

Simplifica $R_{5}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 2 & - 2 & 0 & - 10 & - 10 \\ 0 & - 2 & 3 & 1 & - 7 & 1 \\ 0 & 3 & - 3 & 0 & 2 & 24 \end{array}]$

Paso 5

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

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Paso 5.1

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

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & - 2 - 2 \cdot - 1 & 0 - 2 \cdot 0 & - 10 - 2 \cdot - 2 & - 10 - 2 \cdot 1 \\ 0 & - 2 & 3 & 1 & - 7 & 1 \\ 0 & 3 & - 3 & 0 & 2 & 24 \end{array}]$

Paso 5.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 0 & 0 & - 6 & - 12 \\ 0 & - 2 & 3 & 1 & - 7 & 1 \\ 0 & 3 & - 3 & 0 & 2 & 24 \end{array}]$

Paso 6

Perform the row operation $R_{4} = R_{4} + 2 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Paso 6.1

Perform the row operation $R_{4} = R_{4} + 2 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 0 & 0 & - 6 & - 12 \\ 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 3 + 2 \cdot - 1 & 1 + 2 \cdot 0 & - 7 + 2 \cdot - 2 & 1 + 2 \cdot 1 \\ 0 & 3 & - 3 & 0 & 2 & 24 \end{array}]$

Paso 6.2

Simplifica $R_{4}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 0 & 0 & - 6 & - 12 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 3 & - 3 & 0 & 2 & 24 \end{array}]$

Paso 7

Perform the row operation $R_{5} = R_{5} - 3 R_{2}$ to make the entry at $5, 2$ a $0$ .

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Paso 7.1

Perform the row operation $R_{5} = R_{5} - 3 R_{2}$ to make the entry at $5, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 0 & 0 & - 6 & - 12 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 - 3 \cdot 0 & 3 - 3 \cdot 1 & - 3 - 3 \cdot - 1 & 0 - 3 \cdot 0 & 2 - 3 \cdot - 2 & 24 - 3 \cdot 1 \end{array}]$

Paso 7.2

Simplifica $R_{5}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 0 & 0 & - 6 & - 12 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 0 & 0 & 0 & 8 & 21 \end{array}]$

Paso 8

Swap $R_{4}$ with $R_{3}$ to put a nonzero entry at $3, 3$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 0 & 0 & 0 & - 6 & - 12 \\ 0 & 0 & 0 & 0 & 8 & 21 \end{array}]$

Paso 9

Multiply each element of $R_{4}$ by $- \frac{1}{6}$ to make the entry at $4, 5$ a $1$ .

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Paso 9.1

Multiply each element of $R_{4}$ by $- \frac{1}{6}$ to make the entry at $4, 5$ a $1$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot - 6 & - \frac{1}{6} \cdot - 12 \\ 0 & 0 & 0 & 0 & 8 & 21 \end{array}]$

Paso 9.2

Simplifica $R_{4}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 8 & 21 \end{array}]$

Paso 10

Perform the row operation $R_{5} = R_{5} - 8 R_{4}$ to make the entry at $5, 5$ a $0$ .

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Paso 10.1

Perform the row operation $R_{5} = R_{5} - 8 R_{4}$ to make the entry at $5, 5$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 - 8 \cdot 0 & 0 - 8 \cdot 0 & 0 - 8 \cdot 0 & 0 - 8 \cdot 0 & 8 - 8 \cdot 1 & 21 - 8 \cdot 2 \end{array}]$

Paso 10.2

Simplifica $R_{5}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 5 \end{array}]$

Paso 11

Multiply each element of $R_{5}$ by $\frac{1}{5}$ to make the entry at $5, 6$ a $1$ .

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Paso 11.1

Multiply each element of $R_{5}$ by $\frac{1}{5}$ to make the entry at $5, 6$ a $1$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ \frac{0}{5} & \frac{0}{5} & \frac{0}{5} & \frac{0}{5} & \frac{0}{5} & \frac{5}{5} \end{array}]$

Paso 11.2

Simplifica $R_{5}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Paso 12

Perform the row operation $R_{4} = R_{4} - 2 R_{5}$ to make the entry at $4, 6$ a $0$ .

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Paso 12.1

Perform the row operation $R_{4} = R_{4} - 2 R_{5}$ to make the entry at $4, 6$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & 2 - 2 \cdot 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Paso 12.2

Simplifica $R_{4}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 3 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Paso 13

Perform the row operation $R_{3} = R_{3} - 3 R_{5}$ to make the entry at $3, 6$ a $0$ .

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Paso 13.1

Perform the row operation $R_{3} = R_{3} - 3 R_{5}$ to make the entry at $3, 6$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 - 3 \cdot 0 & 0 - 3 \cdot 0 & 1 - 3 \cdot 0 & 1 - 3 \cdot 0 & - 11 - 3 \cdot 0 & 3 - 3 \cdot 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Paso 13.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 1 \\ 0 & 0 & 1 & 1 & - 11 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Paso 14

Perform the row operation $R_{2} = R_{2} - R_{5}$ to make the entry at $2, 6$ a $0$ .

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Paso 14.1

Perform the row operation $R_{2} = R_{2} - R_{5}$ to make the entry at $2, 6$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 - 0 & 1 - 0 & - 1 - 0 & 0 - 0 & - 2 - 0 & 1 - 1 \\ 0 & 0 & 1 & 1 & - 11 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Paso 14.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & - 1 \\ 0 & 1 & - 1 & 0 & - 2 & 0 \\ 0 & 0 & 1 & 1 & - 11 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Paso 15

Perform the row operation $R_{1} = R_{1} + R_{5}$ to make the entry at $1, 6$ a $0$ .

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Paso 15.1

Perform the row operation $R_{1} = R_{1} + R_{5}$ to make the entry at $1, 6$ a $0$ .

$[\begin{array}{c} 1 + 0 & 1 + 0 & - 2 + 0 & 0 + 0 & - 3 + 0 & - 1 + 1 \cdot 1 \\ 0 & 1 & - 1 & 0 & - 2 & 0 \\ 0 & 0 & 1 & 1 & - 11 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Paso 15.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & 0 \\ 0 & 1 & - 1 & 0 & - 2 & 0 \\ 0 & 0 & 1 & 1 & - 11 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Paso 16

Perform the row operation $R_{3} = R_{3} + 11 R_{4}$ to make the entry at $3, 5$ a $0$ .

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Paso 16.1

Perform the row operation $R_{3} = R_{3} + 11 R_{4}$ to make the entry at $3, 5$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 2 & 0 & - 3 & 0 \\ 0 & 1 & - 1 & 0 & - 2 & 0 \\ 0 + 11 \cdot 0 & 0 + 11 \cdot 0 & 1 + 11 \cdot 0 & 1 + 11 \cdot 0 & - 11 + 11 \cdot 1 & 0 + 11 \cdot 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Paso 16.2

Simplifica $R_{3}$ .

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

Paso 17

Perform the row operation $R_{2} = R_{2} + 2 R_{4}$ to make the entry at $2, 5$ a $0$ .

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Paso 17.1

Perform the row operation $R_{2} = R_{2} + 2 R_{4}$ to make the entry at $2, 5$ a $0$ .

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

Paso 17.2

Simplifica $R_{2}$ .

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

Paso 18

Perform the row operation $R_{1} = R_{1} + 3 R_{4}$ to make the entry at $1, 5$ a $0$ .

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Paso 18.1

Perform the row operation $R_{1} = R_{1} + 3 R_{4}$ to make the entry at $1, 5$ a $0$ .

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

Paso 18.2

Simplifica $R_{1}$ .

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

Paso 19

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

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Paso 19.1

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

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

Paso 19.2

Simplifica $R_{2}$ .

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

Paso 20

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

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Paso 20.1

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

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

Paso 20.2

Simplifica $R_{1}$ .

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

Paso 21

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

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Paso 21.1

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

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

Paso 21.2

Simplifica $R_{1}$ .

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