Álgebra lineal Ejemplos

$[\begin{array}{c} 3 & 6 & - 5 & - 1 & - 8 & x \\ 6 & - 2 & 3 & 2 & 13 & y \\ 4 & - 3 & - 1 & - 3 & - 1 & z \\ 5 & 6 & - 3 & 4 & - 6 & t \end{array}]$

Paso 1

Obtén la forma escalonada reducida por filas.

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Paso 1.1

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

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Paso 1.1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{6}{3} & \frac{- 5}{3} & \frac{- 1}{3} & \frac{- 8}{3} & \frac{x}{3} \\ 6 & - 2 & 3 & 2 & 13 & y \\ 4 & - 3 & - 1 & - 3 & - 1 & z \\ 5 & 6 & - 3 & 4 & - 6 & t \end{array}]$

Paso 1.1.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & - \frac{1}{3} & - \frac{8}{3} & \frac{x}{3} \\ 6 & - 2 & 3 & 2 & 13 & y \\ 4 & - 3 & - 1 & - 3 & - 1 & z \\ 5 & 6 & - 3 & 4 & - 6 & t \end{array}]$

Paso 1.2

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Paso 1.2.1

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & - \frac{1}{3} & - \frac{8}{3} & \frac{x}{3} \\ 6 - 6 \cdot 1 & - 2 - 6 \cdot 2 & 3 - 6 (- \frac{5}{3}) & 2 - 6 (- \frac{1}{3}) & 13 - 6 (- \frac{8}{3}) & y - 6 \frac{x}{3} \\ 4 & - 3 & - 1 & - 3 & - 1 & z \\ 5 & 6 & - 3 & 4 & - 6 & t \end{array}]$

Paso 1.2.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & - \frac{1}{3} & - \frac{8}{3} & \frac{x}{3} \\ 0 & - 14 & 13 & 4 & 29 & y - 2 x \\ 4 & - 3 & - 1 & - 3 & - 1 & z \\ 5 & 6 & - 3 & 4 & - 6 & t \end{array}]$

Paso 1.3

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

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Paso 1.3.1

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

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & - \frac{1}{3} & - \frac{8}{3} & \frac{x}{3} \\ 0 & - 14 & 13 & 4 & 29 & y - 2 x \\ 4 - 4 \cdot 1 & - 3 - 4 \cdot 2 & - 1 - 4 (- \frac{5}{3}) & - 3 - 4 (- \frac{1}{3}) & - 1 - 4 (- \frac{8}{3}) & z - 4 \frac{x}{3} \\ 5 & 6 & - 3 & 4 & - 6 & t \end{array}]$

Paso 1.3.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & - \frac{1}{3} & - \frac{8}{3} & \frac{x}{3} \\ 0 & - 14 & 13 & 4 & 29 & y - 2 x \\ 0 & - 11 & \frac{17}{3} & - \frac{5}{3} & \frac{29}{3} & z - \frac{4 x}{3} \\ 5 & 6 & - 3 & 4 & - 6 & t \end{array}]$

Paso 1.4

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

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Paso 1.4.1

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

Paso 1.4.2

Simplifica $R_{4}$ .

Paso 1.5

Multiply each element of $R_{2}$ by $- \frac{1}{14}$ to make the entry at $2, 2$ a $1$ .

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Paso 1.5.1

Multiply each element of $R_{2}$ by $- \frac{1}{14}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & - \frac{1}{3} & - \frac{8}{3} & \frac{x}{3} \\ - \frac{1}{14} \cdot 0 & - \frac{1}{14} \cdot - 14 & - \frac{1}{14} \cdot 13 & - \frac{1}{14} \cdot 4 & - \frac{1}{14} \cdot 29 & - \frac{1}{14} (y - 2 x) \\ 0 & - 11 & \frac{17}{3} & - \frac{5}{3} & \frac{29}{3} & z - \frac{4 x}{3} \\ 0 & - 4 & \frac{16}{3} & \frac{17}{3} & \frac{22}{3} & t - \frac{5 x}{3} \end{array}]$

Paso 1.5.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & - \frac{1}{3} & - \frac{8}{3} & \frac{x}{3} \\ 0 & 1 & - \frac{13}{14} & - \frac{2}{7} & - \frac{29}{14} & - \frac{y}{14} + \frac{x}{7} \\ 0 & - 11 & \frac{17}{3} & - \frac{5}{3} & \frac{29}{3} & z - \frac{4 x}{3} \\ 0 & - 4 & \frac{16}{3} & \frac{17}{3} & \frac{22}{3} & t - \frac{5 x}{3} \end{array}]$

Paso 1.6

Perform the row operation $R_{3} = R_{3} + 11 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Paso 1.6.1

Perform the row operation $R_{3} = R_{3} + 11 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & - \frac{1}{3} & - \frac{8}{3} & \frac{x}{3} \\ 0 & 1 & - \frac{13}{14} & - \frac{2}{7} & - \frac{29}{14} & - \frac{y}{14} + \frac{x}{7} \\ 0 + 11 \cdot 0 & - 11 + 11 \cdot 1 & \frac{17}{3} + 11 (- \frac{13}{14}) & - \frac{5}{3} + 11 (- \frac{2}{7}) & \frac{29}{3} + 11 (- \frac{29}{14}) & z - \frac{4 x}{3} + 11 (- \frac{y}{14} + \frac{x}{7}) \\ 0 & - 4 & \frac{16}{3} & \frac{17}{3} & \frac{22}{3} & t - \frac{5 x}{3} \end{array}]$

Paso 1.6.2

Simplifica $R_{3}$ .

Paso 1.7

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

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Paso 1.7.1

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

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & - \frac{1}{3} & - \frac{8}{3} & \frac{x}{3} \\ 0 & 1 & - \frac{13}{14} & - \frac{2}{7} & - \frac{29}{14} & - \frac{y}{14} + \frac{x}{7} \\ 0 & 0 & - \frac{191}{42} & - \frac{101}{21} & - \frac{551}{42} & z - \frac{11 y}{14} + \frac{5 x}{21} \\ 0 + 4 \cdot 0 & - 4 + 4 \cdot 1 & \frac{16}{3} + 4 (- \frac{13}{14}) & \frac{17}{3} + 4 (- \frac{2}{7}) & \frac{22}{3} + 4 (- \frac{29}{14}) & t - \frac{5 x}{3} + 4 (- \frac{y}{14} + \frac{x}{7}) \end{array}]$

Paso 1.7.2

Simplifica $R_{4}$ .

Paso 1.8

Multiply each element of $R_{3}$ by $- \frac{42}{191}$ to make the entry at $3, 3$ a $1$ .

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Paso 1.8.1

Multiply each element of $R_{3}$ by $- \frac{42}{191}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & - \frac{1}{3} & - \frac{8}{3} & \frac{x}{3} \\ 0 & 1 & - \frac{13}{14} & - \frac{2}{7} & - \frac{29}{14} & - \frac{y}{14} + \frac{x}{7} \\ - \frac{42}{191} \cdot 0 & - \frac{42}{191} \cdot 0 & - \frac{42}{191} (- \frac{191}{42}) & - \frac{42}{191} (- \frac{101}{21}) & - \frac{42}{191} (- \frac{551}{42}) & - \frac{42}{191} (z - \frac{11 y}{14} + \frac{5 x}{21}) \\ 0 & 0 & \frac{34}{21} & \frac{95}{21} & - \frac{20}{21} & t - \frac{2 y}{7} - \frac{23 x}{21} \end{array}]$

Paso 1.8.2

Simplifica $R_{3}$ .

Paso 1.9

Perform the row operation $R_{4} = R_{4} - \frac{34}{21} R_{3}$ to make the entry at $4, 3$ a $0$ .

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Paso 1.9.1

Perform the row operation $R_{4} = R_{4} - \frac{34}{21} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & - \frac{1}{3} & - \frac{8}{3} & \frac{x}{3} \\ 0 & 1 & - \frac{13}{14} & - \frac{2}{7} & - \frac{29}{14} & - \frac{y}{14} + \frac{x}{7} \\ 0 & 0 & 1 & \frac{202}{191} & \frac{551}{191} & - \frac{42 z}{191} + \frac{33 y}{191} - \frac{10 x}{191} \\ 0 - \frac{34}{21} \cdot 0 & 0 - \frac{34}{21} \cdot 0 & \frac{34}{21} - \frac{34}{21} \cdot 1 & \frac{95}{21} - \frac{34}{21} \cdot \frac{202}{191} & - \frac{20}{21} - \frac{34}{21} \cdot \frac{551}{191} & t - \frac{2 y}{7} - \frac{23 x}{21} - \frac{34}{21} (- \frac{42 z}{191} + \frac{33 y}{191} - \frac{10 x}{191}) \end{array}]$

Paso 1.9.2

Simplifica $R_{4}$ .

Paso 1.10

Multiply each element of $R_{4}$ by $\frac{191}{537}$ to make the entry at $4, 4$ a $1$ .

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Paso 1.10.1

Multiply each element of $R_{4}$ by $\frac{191}{537}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & - \frac{1}{3} & - \frac{8}{3} & \frac{x}{3} \\ 0 & 1 & - \frac{13}{14} & - \frac{2}{7} & - \frac{29}{14} & - \frac{y}{14} + \frac{x}{7} \\ 0 & 0 & 1 & \frac{202}{191} & \frac{551}{191} & - \frac{42 z}{191} + \frac{33 y}{191} - \frac{10 x}{191} \\ \frac{191}{537} \cdot 0 & \frac{191}{537} \cdot 0 & \frac{191}{537} \cdot 0 & \frac{191}{537} \cdot \frac{537}{191} & \frac{191}{537} (- \frac{1074}{191}) & \frac{191}{537} (t + \frac{68 z}{191} - \frac{108 y}{191} - \frac{193 x}{191}) \end{array}]$

Paso 1.10.2

Simplifica $R_{4}$ .

Paso 1.11

Perform the row operation $R_{3} = R_{3} - \frac{202}{191} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Paso 1.11.1

Perform the row operation $R_{3} = R_{3} - \frac{202}{191} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & - \frac{1}{3} & - \frac{8}{3} & \frac{x}{3} \\ 0 & 1 & - \frac{13}{14} & - \frac{2}{7} & - \frac{29}{14} & - \frac{y}{14} + \frac{x}{7} \\ 0 - \frac{202}{191} \cdot 0 & 0 - \frac{202}{191} \cdot 0 & 1 - \frac{202}{191} \cdot 0 & \frac{202}{191} - \frac{202}{191} \cdot 1 & \frac{551}{191} - \frac{202}{191} \cdot - 2 & - \frac{42 z}{191} + \frac{33 y}{191} - \frac{10 x}{191} - \frac{202}{191} (\frac{191 t}{537} + \frac{68 z}{537} - \frac{36 y}{179} - \frac{193 x}{537}) \\ 0 & 0 & 0 & 1 & - 2 & \frac{191 t}{537} + \frac{68 z}{537} - \frac{36 y}{179} - \frac{193 x}{537} \end{array}]$

Paso 1.11.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & - \frac{1}{3} & - \frac{8}{3} & \frac{x}{3} \\ 0 & 1 & - \frac{13}{14} & - \frac{2}{7} & - \frac{29}{14} & - \frac{y}{14} + \frac{x}{7} \\ 0 & 0 & 1 & 0 & 5 & - \frac{202 t - 176 x - 207 y + 190 z}{537} \\ 0 & 0 & 0 & 1 & - 2 & \frac{191 t}{537} + \frac{68 z}{537} - \frac{36 y}{179} - \frac{193 x}{537} \end{array}]$

Paso 1.12

Perform the row operation $R_{2} = R_{2} + \frac{2}{7} R_{4}$ to make the entry at $2, 4$ a $0$ .

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Paso 1.12.1

Perform the row operation $R_{2} = R_{2} + \frac{2}{7} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & - \frac{1}{3} & - \frac{8}{3} & \frac{x}{3} \\ 0 + \frac{2}{7} \cdot 0 & 1 + \frac{2}{7} \cdot 0 & - \frac{13}{14} + \frac{2}{7} \cdot 0 & - \frac{2}{7} + \frac{2}{7} \cdot 1 & - \frac{29}{14} + \frac{2}{7} \cdot - 2 & - \frac{y}{14} + \frac{x}{7} + \frac{2}{7} (\frac{191 t}{537} + \frac{68 z}{537} - \frac{36 y}{179} - \frac{193 x}{537}) \\ 0 & 0 & 1 & 0 & 5 & - \frac{202 t - 176 x - 207 y + 190 z}{537} \\ 0 & 0 & 0 & 1 & - 2 & \frac{191 t}{537} + \frac{68 z}{537} - \frac{36 y}{179} - \frac{193 x}{537} \end{array}]$

Paso 1.12.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & - \frac{1}{3} & - \frac{8}{3} & \frac{x}{3} \\ 0 & 1 & - \frac{13}{14} & 0 & - \frac{37}{14} & \frac{764 t + 272 z + 302 x - 969 y}{7518} \\ 0 & 0 & 1 & 0 & 5 & - \frac{202 t - 176 x - 207 y + 190 z}{537} \\ 0 & 0 & 0 & 1 & - 2 & \frac{191 t}{537} + \frac{68 z}{537} - \frac{36 y}{179} - \frac{193 x}{537} \end{array}]$

Paso 1.13

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

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Paso 1.13.1

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

$[\begin{array}{c} 1 + \frac{1}{3} \cdot 0 & 2 + \frac{1}{3} \cdot 0 & - \frac{5}{3} + \frac{1}{3} \cdot 0 & - \frac{1}{3} + \frac{1}{3} \cdot 1 & - \frac{8}{3} + \frac{1}{3} \cdot - 2 & \frac{x}{3} + \frac{1}{3} (\frac{191 t}{537} + \frac{68 z}{537} - \frac{36 y}{179} - \frac{193 x}{537}) \\ 0 & 1 & - \frac{13}{14} & 0 & - \frac{37}{14} & \frac{764 t + 272 z + 302 x - 969 y}{7518} \\ 0 & 0 & 1 & 0 & 5 & - \frac{202 t - 176 x - 207 y + 190 z}{537} \\ 0 & 0 & 0 & 1 & - 2 & \frac{191 t}{537} + \frac{68 z}{537} - \frac{36 y}{179} - \frac{193 x}{537} \end{array}]$

Paso 1.13.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & 0 & - \frac{10}{3} & \frac{191 t + 68 z + 344 x - 108 y}{1611} \\ 0 & 1 & - \frac{13}{14} & 0 & - \frac{37}{14} & \frac{764 t + 272 z + 302 x - 969 y}{7518} \\ 0 & 0 & 1 & 0 & 5 & - \frac{202 t - 176 x - 207 y + 190 z}{537} \\ 0 & 0 & 0 & 1 & - 2 & \frac{191 t}{537} + \frac{68 z}{537} - \frac{36 y}{179} - \frac{193 x}{537} \end{array}]$

Paso 1.14

Perform the row operation $R_{2} = R_{2} + \frac{13}{14} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Paso 1.14.1

Perform the row operation $R_{2} = R_{2} + \frac{13}{14} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & 0 & - \frac{10}{3} & \frac{191 t + 68 z + 344 x - 108 y}{1611} \\ 0 + \frac{13}{14} \cdot 0 & 1 + \frac{13}{14} \cdot 0 & - \frac{13}{14} + \frac{13}{14} \cdot 1 & 0 + \frac{13}{14} \cdot 0 & - \frac{37}{14} + \frac{13}{14} \cdot 5 & \frac{764 t + 272 z + 302 x - 969 y}{7518} + \frac{13}{14} (- \frac{202 t - 176 x - 207 y + 190 z}{537}) \\ 0 & 0 & 1 & 0 & 5 & - \frac{202 t - 176 x - 207 y + 190 z}{537} \\ 0 & 0 & 0 & 1 & - 2 & \frac{191 t}{537} + \frac{68 z}{537} - \frac{36 y}{179} - \frac{193 x}{537} \end{array}]$

Paso 1.14.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 2 & - \frac{5}{3} & 0 & - \frac{10}{3} & \frac{191 t + 68 z + 344 x - 108 y}{1611} \\ 0 & 1 & 0 & 0 & 2 & - \frac{133 t - 185 x - 123 y + 157 z}{537} \\ 0 & 0 & 1 & 0 & 5 & - \frac{202 t - 176 x - 207 y + 190 z}{537} \\ 0 & 0 & 0 & 1 & - 2 & \frac{191 t}{537} + \frac{68 z}{537} - \frac{36 y}{179} - \frac{193 x}{537} \end{array}]$

Paso 1.15

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

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Paso 1.15.1

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

$[\begin{array}{c} 1 + \frac{5}{3} \cdot 0 & 2 + \frac{5}{3} \cdot 0 & - \frac{5}{3} + \frac{5}{3} \cdot 1 & 0 + \frac{5}{3} \cdot 0 & - \frac{10}{3} + \frac{5}{3} \cdot 5 & \frac{191 t + 68 z + 344 x - 108 y}{1611} + \frac{5}{3} (- \frac{202 t - 176 x - 207 y + 190 z}{537}) \\ 0 & 1 & 0 & 0 & 2 & - \frac{133 t - 185 x - 123 y + 157 z}{537} \\ 0 & 0 & 1 & 0 & 5 & - \frac{202 t - 176 x - 207 y + 190 z}{537} \\ 0 & 0 & 0 & 1 & - 2 & \frac{191 t}{537} + \frac{68 z}{537} - \frac{36 y}{179} - \frac{193 x}{537} \end{array}]$

Paso 1.15.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 2 & 0 & 0 & 5 & - \frac{91 t - 136 x - 103 y + 98 z}{179} \\ 0 & 1 & 0 & 0 & 2 & - \frac{133 t - 185 x - 123 y + 157 z}{537} \\ 0 & 0 & 1 & 0 & 5 & - \frac{202 t - 176 x - 207 y + 190 z}{537} \\ 0 & 0 & 0 & 1 & - 2 & \frac{191 t}{537} + \frac{68 z}{537} - \frac{36 y}{179} - \frac{193 x}{537} \end{array}]$

Paso 1.16

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 1.16.1

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

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 5 - 2 \cdot 2 & - \frac{91 t - 136 x - 103 y + 98 z}{179} - 2 (- \frac{133 t - 185 x - 123 y + 157 z}{537}) \\ 0 & 1 & 0 & 0 & 2 & - \frac{133 t - 185 x - 123 y + 157 z}{537} \\ 0 & 0 & 1 & 0 & 5 & - \frac{202 t - 176 x - 207 y + 190 z}{537} \\ 0 & 0 & 0 & 1 & - 2 & \frac{191 t}{537} + \frac{68 z}{537} - \frac{36 y}{179} - \frac{193 x}{537} \end{array}]$

Paso 1.16.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 1 & - \frac{7 t - 38 x - 63 y - 20 z}{537} \\ 0 & 1 & 0 & 0 & 2 & - \frac{133 t - 185 x - 123 y + 157 z}{537} \\ 0 & 0 & 1 & 0 & 5 & - \frac{202 t - 176 x - 207 y + 190 z}{537} \\ 0 & 0 & 0 & 1 & - 2 & \frac{191 t}{537} + \frac{68 z}{537} - \frac{36 y}{179} - \frac{193 x}{537} \end{array}]$

Paso 2

The pivot positions are the locations with the leading $1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions: $a_{11}, a_{22}, a_{33},$ and $a_{44}$

Pivot Columns: $1, 2, 3,$ and $4$