Algèbre linéaire Exemples

$[\begin{array}{c} 4 & - 1 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 300 \\ - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 & 0 & 0 & 200 \\ 0 & - 1 & 4 & 0 & 0 & - 1 & 0 & 0 & 0 & 300 \\ - 1 & 0 & 0 & 4 & - 1 & 0 & - 1 & 0 & 0 & 100 \\ 0 & - 1 & 0 & - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 \end{array}]$

Étape 1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} 4 & - 1 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 300 & 0 \\ - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 & 0 & 0 & 200 & 0 \\ 0 & - 1 & 4 & 0 & 0 & - 1 & 0 & 0 & 0 & 300 & 0 \\ - 1 & 0 & 0 & 4 & - 1 & 0 & - 1 & 0 & 0 & 100 & 0 \\ 0 & - 1 & 0 & - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2

Déterminez la forme d’échelon en ligne réduite.

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Étape 2.1

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

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Étape 2.1.1

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

$[\begin{array}{c} \frac{4}{4} & \frac{- 1}{4} & \frac{0}{4} & \frac{- 1}{4} & \frac{0}{4} & \frac{0}{4} & \frac{0}{4} & \frac{0}{4} & \frac{0}{4} & \frac{300}{4} & \frac{0}{4} \\ - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 & 0 & 0 & 200 & 0 \\ 0 & - 1 & 4 & 0 & 0 & - 1 & 0 & 0 & 0 & 300 & 0 \\ - 1 & 0 & 0 & 4 & - 1 & 0 & - 1 & 0 & 0 & 100 & 0 \\ 0 & - 1 & 0 & - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.1.2

Simplifiez $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 & 0 & 0 & 200 & 0 \\ 0 & - 1 & 4 & 0 & 0 & - 1 & 0 & 0 & 0 & 300 & 0 \\ - 1 & 0 & 0 & 4 & - 1 & 0 & - 1 & 0 & 0 & 100 & 0 \\ 0 & - 1 & 0 & - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.2

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

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Étape 2.2.1

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

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ - 1 + 1 \cdot 1 & 4 - \frac{1}{4} & - 1 + 0 & 0 - \frac{1}{4} & - 1 + 0 & 0 + 0 & 0 + 0 & 0 + 0 & 0 + 0 & 200 + 1 \cdot 75 & 0 + 0 \\ 0 & - 1 & 4 & 0 & 0 & - 1 & 0 & 0 & 0 & 300 & 0 \\ - 1 & 0 & 0 & 4 & - 1 & 0 & - 1 & 0 & 0 & 100 & 0 \\ 0 & - 1 & 0 & - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.2.2

Simplifiez $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & \frac{15}{4} & - 1 & - \frac{1}{4} & - 1 & 0 & 0 & 0 & 0 & 275 & 0 \\ 0 & - 1 & 4 & 0 & 0 & - 1 & 0 & 0 & 0 & 300 & 0 \\ - 1 & 0 & 0 & 4 & - 1 & 0 & - 1 & 0 & 0 & 100 & 0 \\ 0 & - 1 & 0 & - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.3

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

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Étape 2.3.1

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

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & \frac{15}{4} & - 1 & - \frac{1}{4} & - 1 & 0 & 0 & 0 & 0 & 275 & 0 \\ 0 & - 1 & 4 & 0 & 0 & - 1 & 0 & 0 & 0 & 300 & 0 \\ - 1 + 1 \cdot 1 & 0 - \frac{1}{4} & 0 + 0 & 4 - \frac{1}{4} & - 1 + 0 & 0 + 0 & - 1 + 0 & 0 + 0 & 0 + 0 & 100 + 1 \cdot 75 & 0 + 0 \\ 0 & - 1 & 0 & - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.3.2

Simplifiez $R_{4}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & \frac{15}{4} & - 1 & - \frac{1}{4} & - 1 & 0 & 0 & 0 & 0 & 275 & 0 \\ 0 & - 1 & 4 & 0 & 0 & - 1 & 0 & 0 & 0 & 300 & 0 \\ 0 & - \frac{1}{4} & 0 & \frac{15}{4} & - 1 & 0 & - 1 & 0 & 0 & 175 & 0 \\ 0 & - 1 & 0 & - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.4

Multiply each element of $R_{2}$ by $\frac{4}{15}$ to make the entry at $2, 2$ a $1$ .

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Étape 2.4.1

Multiply each element of $R_{2}$ by $\frac{4}{15}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ \frac{4}{15} \cdot 0 & \frac{4}{15} \cdot \frac{15}{4} & \frac{4}{15} \cdot - 1 & \frac{4}{15} (- \frac{1}{4}) & \frac{4}{15} \cdot - 1 & \frac{4}{15} \cdot 0 & \frac{4}{15} \cdot 0 & \frac{4}{15} \cdot 0 & \frac{4}{15} \cdot 0 & \frac{4}{15} \cdot 275 & \frac{4}{15} \cdot 0 \\ 0 & - 1 & 4 & 0 & 0 & - 1 & 0 & 0 & 0 & 300 & 0 \\ 0 & - \frac{1}{4} & 0 & \frac{15}{4} & - 1 & 0 & - 1 & 0 & 0 & 175 & 0 \\ 0 & - 1 & 0 & - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.4.2

Simplifiez $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & - 1 & 4 & 0 & 0 & - 1 & 0 & 0 & 0 & 300 & 0 \\ 0 & - \frac{1}{4} & 0 & \frac{15}{4} & - 1 & 0 & - 1 & 0 & 0 & 175 & 0 \\ 0 & - 1 & 0 & - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.5

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

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Étape 2.5.1

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

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 + 0 & - 1 + 1 \cdot 1 & 4 - \frac{4}{15} & 0 - \frac{1}{15} & 0 - \frac{4}{15} & - 1 + 0 & 0 + 0 & 0 + 0 & 0 + 0 & 300 + \frac{220}{3} & 0 + 0 \\ 0 & - \frac{1}{4} & 0 & \frac{15}{4} & - 1 & 0 & - 1 & 0 & 0 & 175 & 0 \\ 0 & - 1 & 0 & - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.5.2

Simplifiez $R_{3}$ .

Étape 2.6

Perform the row operation $R_{4} = R_{4} + \frac{1}{4} R_{2}$ to make the entry at $4, 2$ a $0$ .

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Étape 2.6.1

Perform the row operation $R_{4} = R_{4} + \frac{1}{4} R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & \frac{56}{15} & - \frac{1}{15} & - \frac{4}{15} & - 1 & 0 & 0 & 0 & \frac{1120}{3} & 0 \\ 0 + \frac{1}{4} \cdot 0 & - \frac{1}{4} + \frac{1}{4} \cdot 1 & 0 + \frac{1}{4} (- \frac{4}{15}) & \frac{15}{4} + \frac{1}{4} (- \frac{1}{15}) & - 1 + \frac{1}{4} (- \frac{4}{15}) & 0 + \frac{1}{4} \cdot 0 & - 1 + \frac{1}{4} \cdot 0 & 0 + \frac{1}{4} \cdot 0 & 0 + \frac{1}{4} \cdot 0 & 175 + \frac{1}{4} \cdot \frac{220}{3} & 0 + \frac{1}{4} \cdot 0 \\ 0 & - 1 & 0 & - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.6.2

Simplifiez $R_{4}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & \frac{56}{15} & - \frac{1}{15} & - \frac{4}{15} & - 1 & 0 & 0 & 0 & \frac{1120}{3} & 0 \\ 0 & 0 & - \frac{1}{15} & \frac{56}{15} & - \frac{16}{15} & 0 & - 1 & 0 & 0 & \frac{580}{3} & 0 \\ 0 & - 1 & 0 & - 1 & 4 & - 1 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.7

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

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Étape 2.7.1

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

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & \frac{56}{15} & - \frac{1}{15} & - \frac{4}{15} & - 1 & 0 & 0 & 0 & \frac{1120}{3} & 0 \\ 0 & 0 & - \frac{1}{15} & \frac{56}{15} & - \frac{16}{15} & 0 & - 1 & 0 & 0 & \frac{580}{3} & 0 \\ 0 + 0 & - 1 + 1 \cdot 1 & 0 - \frac{4}{15} & - 1 - \frac{1}{15} & 4 - \frac{4}{15} & - 1 + 0 & 0 + 0 & - 1 + 0 & 0 + 0 & 0 + \frac{220}{3} & 0 + 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.7.2

Simplifiez $R_{5}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & \frac{56}{15} & - \frac{1}{15} & - \frac{4}{15} & - 1 & 0 & 0 & 0 & \frac{1120}{3} & 0 \\ 0 & 0 & - \frac{1}{15} & \frac{56}{15} & - \frac{16}{15} & 0 & - 1 & 0 & 0 & \frac{580}{3} & 0 \\ 0 & 0 & - \frac{4}{15} & - \frac{16}{15} & \frac{56}{15} & - 1 & 0 & - 1 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.8

Multiply each element of $R_{3}$ by $\frac{15}{56}$ to make the entry at $3, 3$ a $1$ .

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Étape 2.8.1

Multiply each element of $R_{3}$ by $\frac{15}{56}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ \frac{15}{56} \cdot 0 & \frac{15}{56} \cdot 0 & \frac{15}{56} \cdot \frac{56}{15} & \frac{15}{56} (- \frac{1}{15}) & \frac{15}{56} (- \frac{4}{15}) & \frac{15}{56} \cdot - 1 & \frac{15}{56} \cdot 0 & \frac{15}{56} \cdot 0 & \frac{15}{56} \cdot 0 & \frac{15}{56} \cdot \frac{1120}{3} & \frac{15}{56} \cdot 0 \\ 0 & 0 & - \frac{1}{15} & \frac{56}{15} & - \frac{16}{15} & 0 & - 1 & 0 & 0 & \frac{580}{3} & 0 \\ 0 & 0 & - \frac{4}{15} & - \frac{16}{15} & \frac{56}{15} & - 1 & 0 & - 1 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.8.2

Simplifiez $R_{3}$ .

Étape 2.9

Perform the row operation $R_{4} = R_{4} + \frac{1}{15} R_{3}$ to make the entry at $4, 3$ a $0$ .

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Étape 2.9.1

Perform the row operation $R_{4} = R_{4} + \frac{1}{15} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 + \frac{1}{15} \cdot 0 & 0 + \frac{1}{15} \cdot 0 & - \frac{1}{15} + \frac{1}{15} \cdot 1 & \frac{56}{15} + \frac{1}{15} (- \frac{1}{56}) & - \frac{16}{15} + \frac{1}{15} (- \frac{1}{14}) & 0 + \frac{1}{15} (- \frac{15}{56}) & - 1 + \frac{1}{15} \cdot 0 & 0 + \frac{1}{15} \cdot 0 & 0 + \frac{1}{15} \cdot 0 & \frac{580}{3} + \frac{1}{15} \cdot 100 & 0 + \frac{1}{15} \cdot 0 \\ 0 & 0 & - \frac{4}{15} & - \frac{16}{15} & \frac{56}{15} & - 1 & 0 & - 1 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.9.2

Simplifiez $R_{4}$ .

Étape 2.10

Perform the row operation $R_{5} = R_{5} + \frac{4}{15} R_{3}$ to make the entry at $5, 3$ a $0$ .

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Étape 2.10.1

Perform the row operation $R_{5} = R_{5} + \frac{4}{15} R_{3}$ to make the entry at $5, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & \frac{209}{56} & - \frac{15}{14} & - \frac{1}{56} & - 1 & 0 & 0 & 200 & 0 \\ 0 + \frac{4}{15} \cdot 0 & 0 + \frac{4}{15} \cdot 0 & - \frac{4}{15} + \frac{4}{15} \cdot 1 & - \frac{16}{15} + \frac{4}{15} (- \frac{1}{56}) & \frac{56}{15} + \frac{4}{15} (- \frac{1}{14}) & - 1 + \frac{4}{15} (- \frac{15}{56}) & 0 + \frac{4}{15} \cdot 0 & - 1 + \frac{4}{15} \cdot 0 & 0 + \frac{4}{15} \cdot 0 & \frac{220}{3} + \frac{4}{15} \cdot 100 & 0 + \frac{4}{15} \cdot 0 \\ 0 & 0 & - 1 & 0 & - 1 & 4 & 0 & 0 & - 1 & 100 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.10.2

Simplifiez $R_{5}$ .

Étape 2.11

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

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Étape 2.11.1

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

Étape 2.11.2

Simplifiez $R_{6}$ .

Étape 2.12

Multiply each element of $R_{4}$ by $\frac{56}{209}$ to make the entry at $4, 4$ a $1$ .

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Étape 2.12.1

Multiply each element of $R_{4}$ by $\frac{56}{209}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ \frac{56}{209} \cdot 0 & \frac{56}{209} \cdot 0 & \frac{56}{209} \cdot 0 & \frac{56}{209} \cdot \frac{209}{56} & \frac{56}{209} (- \frac{15}{14}) & \frac{56}{209} (- \frac{1}{56}) & \frac{56}{209} \cdot - 1 & \frac{56}{209} \cdot 0 & \frac{56}{209} \cdot 0 & \frac{56}{209} \cdot 200 & \frac{56}{209} \cdot 0 \\ 0 & 0 & 0 & - \frac{15}{14} & \frac{26}{7} & - \frac{15}{14} & 0 & - 1 & 0 & 100 & 0 \\ 0 & 0 & 0 & - \frac{1}{56} & - \frac{15}{14} & \frac{209}{56} & 0 & 0 & - 1 & 200 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.12.2

Simplifiez $R_{4}$ .

Étape 2.13

Perform the row operation $R_{5} = R_{5} + \frac{15}{14} R_{4}$ to make the entry at $5, 4$ a $0$ .

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Étape 2.13.1

Perform the row operation $R_{5} = R_{5} + \frac{15}{14} R_{4}$ to make the entry at $5, 4$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 + \frac{15}{14} \cdot 0 & 0 + \frac{15}{14} \cdot 0 & 0 + \frac{15}{14} \cdot 0 & - \frac{15}{14} + \frac{15}{14} \cdot 1 & \frac{26}{7} + \frac{15}{14} (- \frac{60}{209}) & - \frac{15}{14} + \frac{15}{14} (- \frac{1}{209}) & 0 + \frac{15}{14} (- \frac{56}{209}) & - 1 + \frac{15}{14} \cdot 0 & 0 + \frac{15}{14} \cdot 0 & 100 + \frac{15}{14} \cdot \frac{11200}{209} & 0 + \frac{15}{14} \cdot 0 \\ 0 & 0 & 0 & - \frac{1}{56} & - \frac{15}{14} & \frac{209}{56} & 0 & 0 & - 1 & 200 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.13.2

Simplifiez $R_{5}$ .

Étape 2.14

Perform the row operation $R_{6} = R_{6} + \frac{1}{56} R_{4}$ to make the entry at $6, 4$ a $0$ .

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Étape 2.14.1

Perform the row operation $R_{6} = R_{6} + \frac{1}{56} R_{4}$ to make the entry at $6, 4$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & \frac{712}{209} & - \frac{225}{209} & - \frac{60}{209} & - 1 & 0 & \frac{32900}{209} & 0 \\ 0 + \frac{1}{56} \cdot 0 & 0 + \frac{1}{56} \cdot 0 & 0 + \frac{1}{56} \cdot 0 & - \frac{1}{56} + \frac{1}{56} \cdot 1 & - \frac{15}{14} + \frac{1}{56} (- \frac{60}{209}) & \frac{209}{56} + \frac{1}{56} (- \frac{1}{209}) & 0 + \frac{1}{56} (- \frac{56}{209}) & 0 + \frac{1}{56} \cdot 0 & - 1 + \frac{1}{56} \cdot 0 & 200 + \frac{1}{56} \cdot \frac{11200}{209} & 0 + \frac{1}{56} \cdot 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 4 & - 1 & 0 & 160 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.14.2

Simplifiez $R_{6}$ .

Étape 2.15

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

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Étape 2.15.1

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

Étape 2.15.2

Simplifiez $R_{7}$ .

Étape 2.16

Multiply each element of $R_{5}$ by $\frac{209}{712}$ to make the entry at $5, 5$ a $1$ .

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Étape 2.16.1

Multiply each element of $R_{5}$ by $\frac{209}{712}$ to make the entry at $5, 5$ a $1$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ \frac{209}{712} \cdot 0 & \frac{209}{712} \cdot 0 & \frac{209}{712} \cdot 0 & \frac{209}{712} \cdot 0 & \frac{209}{712} \cdot \frac{712}{209} & \frac{209}{712} (- \frac{225}{209}) & \frac{209}{712} (- \frac{60}{209}) & \frac{209}{712} \cdot - 1 & \frac{209}{712} \cdot 0 & \frac{209}{712} \cdot \frac{32900}{209} & \frac{209}{712} \cdot 0 \\ 0 & 0 & 0 & 0 & - \frac{225}{209} & \frac{780}{209} & - \frac{1}{209} & 0 & - 1 & \frac{42000}{209} & 0 \\ 0 & 0 & 0 & 0 & - \frac{60}{209} & - \frac{1}{209} & \frac{780}{209} & - 1 & 0 & \frac{44640}{209} & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.16.2

Simplifiez $R_{5}$ .

Étape 2.17

Perform the row operation $R_{6} = R_{6} + \frac{225}{209} R_{5}$ to make the entry at $6, 5$ a $0$ .

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Étape 2.17.1

Perform the row operation $R_{6} = R_{6} + \frac{225}{209} R_{5}$ to make the entry at $6, 5$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 + \frac{225}{209} \cdot 0 & 0 + \frac{225}{209} \cdot 0 & 0 + \frac{225}{209} \cdot 0 & 0 + \frac{225}{209} \cdot 0 & - \frac{225}{209} + \frac{225}{209} \cdot 1 & \frac{780}{209} + \frac{225}{209} (- \frac{225}{712}) & - \frac{1}{209} + \frac{225}{209} (- \frac{15}{178}) & 0 + \frac{225}{209} (- \frac{209}{712}) & - 1 + \frac{225}{209} \cdot 0 & \frac{42000}{209} + \frac{225}{209} \cdot \frac{8225}{178} & 0 + \frac{225}{209} \cdot 0 \\ 0 & 0 & 0 & 0 & - \frac{60}{209} & - \frac{1}{209} & \frac{780}{209} & - 1 & 0 & \frac{44640}{209} & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.17.2

Simplifiez $R_{6}$ .

Étape 2.18

Perform the row operation $R_{7} = R_{7} + \frac{60}{209} R_{5}$ to make the entry at $7, 5$ a $0$ .

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Étape 2.18.1

Perform the row operation $R_{7} = R_{7} + \frac{60}{209} R_{5}$ to make the entry at $7, 5$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{2415}{712} & - \frac{17}{178} & - \frac{225}{712} & - 1 & \frac{44625}{178} & 0 \\ 0 + \frac{60}{209} \cdot 0 & 0 + \frac{60}{209} \cdot 0 & 0 + \frac{60}{209} \cdot 0 & 0 + \frac{60}{209} \cdot 0 & - \frac{60}{209} + \frac{60}{209} \cdot 1 & - \frac{1}{209} + \frac{60}{209} (- \frac{225}{712}) & \frac{780}{209} + \frac{60}{209} (- \frac{15}{178}) & - 1 + \frac{60}{209} (- \frac{209}{712}) & 0 + \frac{60}{209} \cdot 0 & \frac{44640}{209} + \frac{60}{209} \cdot \frac{8225}{178} & 0 + \frac{60}{209} \cdot 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & - 1 & 60 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.18.2

Simplifiez $R_{7}$ .

Étape 2.19

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

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Étape 2.19.1

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

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{2415}{712} & - \frac{17}{178} & - \frac{225}{712} & - 1 & \frac{44625}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{17}{178} & \frac{330}{89} & - \frac{193}{178} & 0 & \frac{20190}{89} & 0 \\ 0 + 0 & 0 + 0 & 0 + 0 & 0 + 0 & - 1 + 1 \cdot 1 & 0 - \frac{225}{712} & - 1 - \frac{15}{178} & 4 - \frac{209}{712} & - 1 + 0 & 60 + \frac{8225}{178} & 0 + 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.19.2

Simplifiez $R_{8}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{2415}{712} & - \frac{17}{178} & - \frac{225}{712} & - 1 & \frac{44625}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{17}{178} & \frac{330}{89} & - \frac{193}{178} & 0 & \frac{20190}{89} & 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{225}{712} & - \frac{193}{178} & \frac{2639}{712} & - 1 & \frac{18905}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.20

Multiply each element of $R_{6}$ by $\frac{712}{2415}$ to make the entry at $6, 6$ a $1$ .

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Étape 2.20.1

Multiply each element of $R_{6}$ by $\frac{712}{2415}$ to make the entry at $6, 6$ a $1$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ \frac{712}{2415} \cdot 0 & \frac{712}{2415} \cdot 0 & \frac{712}{2415} \cdot 0 & \frac{712}{2415} \cdot 0 & \frac{712}{2415} \cdot 0 & \frac{712}{2415} \cdot \frac{2415}{712} & \frac{712}{2415} (- \frac{17}{178}) & \frac{712}{2415} (- \frac{225}{712}) & \frac{712}{2415} \cdot - 1 & \frac{712}{2415} \cdot \frac{44625}{178} & \frac{712}{2415} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{17}{178} & \frac{330}{89} & - \frac{193}{178} & 0 & \frac{20190}{89} & 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{225}{712} & - \frac{193}{178} & \frac{2639}{712} & - 1 & \frac{18905}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.20.2

Simplifiez $R_{6}$ .

Étape 2.21

Perform the row operation $R_{7} = R_{7} + \frac{17}{178} R_{6}$ to make the entry at $7, 6$ a $0$ .

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Étape 2.21.1

Perform the row operation $R_{7} = R_{7} + \frac{17}{178} R_{6}$ to make the entry at $7, 6$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ 0 + \frac{17}{178} \cdot 0 & 0 + \frac{17}{178} \cdot 0 & 0 + \frac{17}{178} \cdot 0 & 0 + \frac{17}{178} \cdot 0 & 0 + \frac{17}{178} \cdot 0 & - \frac{17}{178} + \frac{17}{178} \cdot 1 & \frac{330}{89} + \frac{17}{178} (- \frac{68}{2415}) & - \frac{193}{178} + \frac{17}{178} (- \frac{15}{161}) & 0 + \frac{17}{178} (- \frac{712}{2415}) & \frac{20190}{89} + \frac{17}{178} \cdot \frac{1700}{23} & 0 + \frac{17}{178} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{225}{712} & - \frac{193}{178} & \frac{2639}{712} & - 1 & \frac{18905}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.21.2

Simplifiez $R_{7}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{8948}{2415} & - \frac{176}{161} & - \frac{68}{2415} & \frac{5380}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{225}{712} & - \frac{193}{178} & \frac{2639}{712} & - 1 & \frac{18905}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.22

Perform the row operation $R_{8} = R_{8} + \frac{225}{712} R_{6}$ to make the entry at $8, 6$ a $0$ .

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Étape 2.22.1

Perform the row operation $R_{8} = R_{8} + \frac{225}{712} R_{6}$ to make the entry at $8, 6$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{8948}{2415} & - \frac{176}{161} & - \frac{68}{2415} & \frac{5380}{23} & 0 \\ 0 + \frac{225}{712} \cdot 0 & 0 + \frac{225}{712} \cdot 0 & 0 + \frac{225}{712} \cdot 0 & 0 + \frac{225}{712} \cdot 0 & 0 + \frac{225}{712} \cdot 0 & - \frac{225}{712} + \frac{225}{712} \cdot 1 & - \frac{193}{178} + \frac{225}{712} (- \frac{68}{2415}) & \frac{2639}{712} + \frac{225}{712} (- \frac{15}{161}) & - 1 + \frac{225}{712} (- \frac{712}{2415}) & \frac{18905}{178} + \frac{225}{712} \cdot \frac{1700}{23} & 0 + \frac{225}{712} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 4 & 160 & 0 \end{array}]$

Étape 2.22.2

Simplifiez $R_{8}$ .

Étape 2.23

Perform the row operation $R_{9} = R_{9} + R_{6}$ to make the entry at $9, 6$ a $0$ .

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Étape 2.23.1

Perform the row operation $R_{9} = R_{9} + R_{6}$ to make the entry at $9, 6$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{8948}{2415} & - \frac{176}{161} & - \frac{68}{2415} & \frac{5380}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - \frac{176}{161} & \frac{592}{161} & - \frac{176}{161} & \frac{2980}{23} & 0 \\ 0 + 0 & 0 + 0 & 0 + 0 & 0 + 0 & 0 + 0 & - 1 + 1 \cdot 1 & 0 - \frac{68}{2415} & - 1 - \frac{15}{161} & 4 - \frac{712}{2415} & 160 + \frac{1700}{23} & 0 + 0 \end{array}]$

Étape 2.23.2

Simplifiez $R_{9}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{8948}{2415} & - \frac{176}{161} & - \frac{68}{2415} & \frac{5380}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - \frac{176}{161} & \frac{592}{161} & - \frac{176}{161} & \frac{2980}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - \frac{68}{2415} & - \frac{176}{161} & \frac{8948}{2415} & \frac{5380}{23} & 0 \end{array}]$

Étape 2.24

Multiply each element of $R_{7}$ by $\frac{2415}{8948}$ to make the entry at $7, 7$ a $1$ .

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Étape 2.24.1

Multiply each element of $R_{7}$ by $\frac{2415}{8948}$ to make the entry at $7, 7$ a $1$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ \frac{2415}{8948} \cdot 0 & \frac{2415}{8948} \cdot 0 & \frac{2415}{8948} \cdot 0 & \frac{2415}{8948} \cdot 0 & \frac{2415}{8948} \cdot 0 & \frac{2415}{8948} \cdot 0 & \frac{2415}{8948} \cdot \frac{8948}{2415} & \frac{2415}{8948} (- \frac{176}{161}) & \frac{2415}{8948} (- \frac{68}{2415}) & \frac{2415}{8948} \cdot \frac{5380}{23} & \frac{2415}{8948} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - \frac{176}{161} & \frac{592}{161} & - \frac{176}{161} & \frac{2980}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - \frac{68}{2415} & - \frac{176}{161} & \frac{8948}{2415} & \frac{5380}{23} & 0 \end{array}]$

Étape 2.24.2

Simplifiez $R_{7}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & - \frac{660}{2237} & - \frac{17}{2237} & \frac{141225}{2237} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - \frac{176}{161} & \frac{592}{161} & - \frac{176}{161} & \frac{2980}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - \frac{68}{2415} & - \frac{176}{161} & \frac{8948}{2415} & \frac{5380}{23} & 0 \end{array}]$

Étape 2.25

Perform the row operation $R_{8} = R_{8} + \frac{176}{161} R_{7}$ to make the entry at $8, 7$ a $0$ .

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Étape 2.25.1

Perform the row operation $R_{8} = R_{8} + \frac{176}{161} R_{7}$ to make the entry at $8, 7$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & - \frac{660}{2237} & - \frac{17}{2237} & \frac{141225}{2237} & 0 \\ 0 + \frac{176}{161} \cdot 0 & 0 + \frac{176}{161} \cdot 0 & 0 + \frac{176}{161} \cdot 0 & 0 + \frac{176}{161} \cdot 0 & 0 + \frac{176}{161} \cdot 0 & 0 + \frac{176}{161} \cdot 0 & - \frac{176}{161} + \frac{176}{161} \cdot 1 & \frac{592}{161} + \frac{176}{161} (- \frac{660}{2237}) & - \frac{176}{161} + \frac{176}{161} (- \frac{17}{2237}) & \frac{2980}{23} + \frac{176}{161} \cdot \frac{141225}{2237} & 0 + \frac{176}{161} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - \frac{68}{2415} & - \frac{176}{161} & \frac{8948}{2415} & \frac{5380}{23} & 0 \end{array}]$

Étape 2.25.2

Simplifiez $R_{8}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & - \frac{660}{2237} & - \frac{17}{2237} & \frac{141225}{2237} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{7504}{2237} & - \frac{2464}{2237} & \frac{444220}{2237} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - \frac{68}{2415} & - \frac{176}{161} & \frac{8948}{2415} & \frac{5380}{23} & 0 \end{array}]$

Étape 2.26

Perform the row operation $R_{9} = R_{9} + \frac{68}{2415} R_{7}$ to make the entry at $9, 7$ a $0$ .

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Étape 2.26.1

Perform the row operation $R_{9} = R_{9} + \frac{68}{2415} R_{7}$ to make the entry at $9, 7$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & - \frac{660}{2237} & - \frac{17}{2237} & \frac{141225}{2237} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{7504}{2237} & - \frac{2464}{2237} & \frac{444220}{2237} & 0 \\ 0 + \frac{68}{2415} \cdot 0 & 0 + \frac{68}{2415} \cdot 0 & 0 + \frac{68}{2415} \cdot 0 & 0 + \frac{68}{2415} \cdot 0 & 0 + \frac{68}{2415} \cdot 0 & 0 + \frac{68}{2415} \cdot 0 & - \frac{68}{2415} + \frac{68}{2415} \cdot 1 & - \frac{176}{161} + \frac{68}{2415} (- \frac{660}{2237}) & \frac{8948}{2415} + \frac{68}{2415} (- \frac{17}{2237}) & \frac{5380}{23} + \frac{68}{2415} \cdot \frac{141225}{2237} & 0 + \frac{68}{2415} \cdot 0 \end{array}]$

Étape 2.26.2

Simplifiez $R_{9}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & - \frac{660}{2237} & - \frac{17}{2237} & \frac{141225}{2237} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{7504}{2237} & - \frac{2464}{2237} & \frac{444220}{2237} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{2464}{2237} & \frac{8288}{2237} & \frac{527240}{2237} & 0 \end{array}]$

Étape 2.27

Multiply each element of $R_{8}$ by $\frac{2237}{7504}$ to make the entry at $8, 8$ a $1$ .

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Étape 2.27.1

Multiply each element of $R_{8}$ by $\frac{2237}{7504}$ to make the entry at $8, 8$ a $1$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & - \frac{660}{2237} & - \frac{17}{2237} & \frac{141225}{2237} & 0 \\ \frac{2237}{7504} \cdot 0 & \frac{2237}{7504} \cdot 0 & \frac{2237}{7504} \cdot 0 & \frac{2237}{7504} \cdot 0 & \frac{2237}{7504} \cdot 0 & \frac{2237}{7504} \cdot 0 & \frac{2237}{7504} \cdot 0 & \frac{2237}{7504} \cdot \frac{7504}{2237} & \frac{2237}{7504} (- \frac{2464}{2237}) & \frac{2237}{7504} \cdot \frac{444220}{2237} & \frac{2237}{7504} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{2464}{2237} & \frac{8288}{2237} & \frac{527240}{2237} & 0 \end{array}]$

Étape 2.27.2

Simplifiez $R_{8}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & - \frac{660}{2237} & - \frac{17}{2237} & \frac{141225}{2237} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & - \frac{22}{67} & \frac{15865}{268} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{2464}{2237} & \frac{8288}{2237} & \frac{527240}{2237} & 0 \end{array}]$

Étape 2.28

Perform the row operation $R_{9} = R_{9} + \frac{2464}{2237} R_{8}$ to make the entry at $9, 8$ a $0$ .

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Étape 2.28.1

Perform the row operation $R_{9} = R_{9} + \frac{2464}{2237} R_{8}$ to make the entry at $9, 8$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & - \frac{660}{2237} & - \frac{17}{2237} & \frac{141225}{2237} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & - \frac{22}{67} & \frac{15865}{268} & 0 \\ 0 + \frac{2464}{2237} \cdot 0 & 0 + \frac{2464}{2237} \cdot 0 & 0 + \frac{2464}{2237} \cdot 0 & 0 + \frac{2464}{2237} \cdot 0 & 0 + \frac{2464}{2237} \cdot 0 & 0 + \frac{2464}{2237} \cdot 0 & 0 + \frac{2464}{2237} \cdot 0 & - \frac{2464}{2237} + \frac{2464}{2237} \cdot 1 & \frac{8288}{2237} + \frac{2464}{2237} (- \frac{22}{67}) & \frac{527240}{2237} + \frac{2464}{2237} \cdot \frac{15865}{268} & 0 + \frac{2464}{2237} \cdot 0 \end{array}]$

Étape 2.28.2

Simplifiez $R_{9}$ .

Étape 2.29

Multiply each element of $R_{9}$ by $\frac{67}{224}$ to make the entry at $9, 9$ a $1$ .

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Étape 2.29.1

Multiply each element of $R_{9}$ by $\frac{67}{224}$ to make the entry at $9, 9$ a $1$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & - \frac{660}{2237} & - \frac{17}{2237} & \frac{141225}{2237} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & - \frac{22}{67} & \frac{15865}{268} & 0 \\ \frac{67}{224} \cdot 0 & \frac{67}{224} \cdot 0 & \frac{67}{224} \cdot 0 & \frac{67}{224} \cdot 0 & \frac{67}{224} \cdot 0 & \frac{67}{224} \cdot 0 & \frac{67}{224} \cdot 0 & \frac{67}{224} \cdot 0 & \frac{67}{224} \cdot \frac{224}{67} & \frac{67}{224} \cdot \frac{20160}{67} & \frac{67}{224} \cdot 0 \end{array}]$

Étape 2.29.2

Simplifiez $R_{9}$ .

Étape 2.30

Perform the row operation $R_{8} = R_{8} + \frac{22}{67} R_{9}$ to make the entry at $8, 9$ a $0$ .

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Étape 2.30.1

Perform the row operation $R_{8} = R_{8} + \frac{22}{67} R_{9}$ to make the entry at $8, 9$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & - \frac{660}{2237} & - \frac{17}{2237} & \frac{141225}{2237} & 0 \\ 0 + \frac{22}{67} \cdot 0 & 0 + \frac{22}{67} \cdot 0 & 0 + \frac{22}{67} \cdot 0 & 0 + \frac{22}{67} \cdot 0 & 0 + \frac{22}{67} \cdot 0 & 0 + \frac{22}{67} \cdot 0 & 0 + \frac{22}{67} \cdot 0 & 1 + \frac{22}{67} \cdot 0 & - \frac{22}{67} + \frac{22}{67} \cdot 1 & \frac{15865}{268} + \frac{22}{67} \cdot 90 & 0 + \frac{22}{67} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.30.2

Simplifiez $R_{8}$ .

Étape 2.31

Perform the row operation $R_{7} = R_{7} + \frac{17}{2237} R_{9}$ to make the entry at $7, 9$ a $0$ .

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Étape 2.31.1

Perform the row operation $R_{7} = R_{7} + \frac{17}{2237} R_{9}$ to make the entry at $7, 9$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & - \frac{712}{2415} & \frac{1700}{23} & 0 \\ 0 + \frac{17}{2237} \cdot 0 & 0 + \frac{17}{2237} \cdot 0 & 0 + \frac{17}{2237} \cdot 0 & 0 + \frac{17}{2237} \cdot 0 & 0 + \frac{17}{2237} \cdot 0 & 0 + \frac{17}{2237} \cdot 0 & 1 + \frac{17}{2237} \cdot 0 & - \frac{660}{2237} + \frac{17}{2237} \cdot 0 & - \frac{17}{2237} + \frac{17}{2237} \cdot 1 & \frac{141225}{2237} + \frac{17}{2237} \cdot 90 & 0 + \frac{17}{2237} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.31.2

Simplifiez $R_{7}$ .

Étape 2.32

Perform the row operation $R_{6} = R_{6} + \frac{712}{2415} R_{9}$ to make the entry at $6, 9$ a $0$ .

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Étape 2.32.1

Perform the row operation $R_{6} = R_{6} + \frac{712}{2415} R_{9}$ to make the entry at $6, 9$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 + \frac{712}{2415} \cdot 0 & 0 + \frac{712}{2415} \cdot 0 & 0 + \frac{712}{2415} \cdot 0 & 0 + \frac{712}{2415} \cdot 0 & 0 + \frac{712}{2415} \cdot 0 & 1 + \frac{712}{2415} \cdot 0 & - \frac{68}{2415} + \frac{712}{2415} \cdot 0 & - \frac{15}{161} + \frac{712}{2415} \cdot 0 & - \frac{712}{2415} + \frac{712}{2415} \cdot 1 & \frac{1700}{23} + \frac{712}{2415} \cdot 90 & 0 + \frac{712}{2415} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & - \frac{660}{2237} & 0 & \frac{142755}{2237} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.32.2

Simplifiez $R_{6}$ .

Étape 2.33

Perform the row operation $R_{7} = R_{7} + \frac{660}{2237} R_{8}$ to make the entry at $7, 8$ a $0$ .

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Étape 2.33.1

Perform the row operation $R_{7} = R_{7} + \frac{660}{2237} R_{8}$ to make the entry at $7, 8$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & - \frac{15}{161} & 0 & \frac{16172}{161} & 0 \\ 0 + \frac{660}{2237} \cdot 0 & 0 + \frac{660}{2237} \cdot 0 & 0 + \frac{660}{2237} \cdot 0 & 0 + \frac{660}{2237} \cdot 0 & 0 + \frac{660}{2237} \cdot 0 & 0 + \frac{660}{2237} \cdot 0 & 1 + \frac{660}{2237} \cdot 0 & - \frac{660}{2237} + \frac{660}{2237} \cdot 1 & 0 + \frac{660}{2237} \cdot 0 & \frac{142755}{2237} + \frac{660}{2237} \cdot \frac{355}{4} & 0 + \frac{660}{2237} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.33.2

Simplifiez $R_{7}$ .

Étape 2.34

Perform the row operation $R_{6} = R_{6} + \frac{15}{161} R_{8}$ to make the entry at $6, 8$ a $0$ .

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Étape 2.34.1

Perform the row operation $R_{6} = R_{6} + \frac{15}{161} R_{8}$ to make the entry at $6, 8$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & - \frac{209}{712} & 0 & \frac{8225}{178} & 0 \\ 0 + \frac{15}{161} \cdot 0 & 0 + \frac{15}{161} \cdot 0 & 0 + \frac{15}{161} \cdot 0 & 0 + \frac{15}{161} \cdot 0 & 0 + \frac{15}{161} \cdot 0 & 1 + \frac{15}{161} \cdot 0 & - \frac{68}{2415} + \frac{15}{161} \cdot 0 & - \frac{15}{161} + \frac{15}{161} \cdot 1 & 0 + \frac{15}{161} \cdot 0 & \frac{16172}{161} + \frac{15}{161} \cdot \frac{355}{4} & 0 + \frac{15}{161} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.34.2

Simplifiez $R_{6}$ .

Étape 2.35

Perform the row operation $R_{5} = R_{5} + \frac{209}{712} R_{8}$ to make the entry at $5, 8$ a $0$ .

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Étape 2.35.1

Perform the row operation $R_{5} = R_{5} + \frac{209}{712} R_{8}$ to make the entry at $5, 8$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 + \frac{209}{712} \cdot 0 & 0 + \frac{209}{712} \cdot 0 & 0 + \frac{209}{712} \cdot 0 & 0 + \frac{209}{712} \cdot 0 & 1 + \frac{209}{712} \cdot 0 & - \frac{225}{712} + \frac{209}{712} \cdot 0 & - \frac{15}{178} + \frac{209}{712} \cdot 0 & - \frac{209}{712} + \frac{209}{712} \cdot 1 & 0 + \frac{209}{712} \cdot 0 & \frac{8225}{178} + \frac{209}{712} \cdot \frac{355}{4} & 0 + \frac{209}{712} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - \frac{68}{2415} & 0 & 0 & \frac{70013}{644} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.35.2

Simplifiez $R_{5}$ .

Étape 2.36

Perform the row operation $R_{6} = R_{6} + \frac{68}{2415} R_{7}$ to make the entry at $6, 7$ a $0$ .

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Étape 2.36.1

Perform the row operation $R_{6} = R_{6} + \frac{68}{2415} R_{7}$ to make the entry at $6, 7$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & - \frac{15}{178} & 0 & 0 & \frac{205795}{2848} & 0 \\ 0 + \frac{68}{2415} \cdot 0 & 0 + \frac{68}{2415} \cdot 0 & 0 + \frac{68}{2415} \cdot 0 & 0 + \frac{68}{2415} \cdot 0 & 0 + \frac{68}{2415} \cdot 0 & 1 + \frac{68}{2415} \cdot 0 & - \frac{68}{2415} + \frac{68}{2415} \cdot 1 & 0 + \frac{68}{2415} \cdot 0 & 0 + \frac{68}{2415} \cdot 0 & \frac{70013}{644} + \frac{68}{2415} \cdot 90 & 0 + \frac{68}{2415} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.36.2

Simplifiez $R_{6}$ .

Étape 2.37

Perform the row operation $R_{5} = R_{5} + \frac{15}{178} R_{7}$ to make the entry at $5, 7$ a $0$ .

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Étape 2.37.1

Perform the row operation $R_{5} = R_{5} + \frac{15}{178} R_{7}$ to make the entry at $5, 7$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & - \frac{56}{209} & 0 & 0 & \frac{11200}{209} & 0 \\ 0 + \frac{15}{178} \cdot 0 & 0 + \frac{15}{178} \cdot 0 & 0 + \frac{15}{178} \cdot 0 & 0 + \frac{15}{178} \cdot 0 & 1 + \frac{15}{178} \cdot 0 & - \frac{225}{712} + \frac{15}{178} \cdot 0 & - \frac{15}{178} + \frac{15}{178} \cdot 1 & 0 + \frac{15}{178} \cdot 0 & 0 + \frac{15}{178} \cdot 0 & \frac{205795}{2848} + \frac{15}{178} \cdot 90 & 0 + \frac{15}{178} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.37.2

Simplifiez $R_{5}$ .

Étape 2.38

Perform the row operation $R_{4} = R_{4} + \frac{56}{209} R_{7}$ to make the entry at $4, 7$ a $0$ .

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Étape 2.38.1

Perform the row operation $R_{4} = R_{4} + \frac{56}{209} R_{7}$ to make the entry at $4, 7$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 + \frac{56}{209} \cdot 0 & 0 + \frac{56}{209} \cdot 0 & 0 + \frac{56}{209} \cdot 0 & 1 + \frac{56}{209} \cdot 0 & - \frac{60}{209} + \frac{56}{209} \cdot 0 & - \frac{1}{209} + \frac{56}{209} \cdot 0 & - \frac{56}{209} + \frac{56}{209} \cdot 1 & 0 + \frac{56}{209} \cdot 0 & 0 + \frac{56}{209} \cdot 0 & \frac{11200}{209} + \frac{56}{209} \cdot 90 & 0 + \frac{56}{209} \cdot 0 \\ 0 & 0 & 0 & 0 & 1 & - \frac{225}{712} & 0 & 0 & 0 & \frac{2555}{32} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.38.2

Simplifiez $R_{4}$ .

Étape 2.39

Perform the row operation $R_{5} = R_{5} + \frac{225}{712} R_{6}$ to make the entry at $5, 6$ a $0$ .

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Étape 2.39.1

Perform the row operation $R_{5} = R_{5} + \frac{225}{712} R_{6}$ to make the entry at $5, 6$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & - \frac{1}{209} & 0 & 0 & 0 & \frac{16240}{209} & 0 \\ 0 + \frac{225}{712} \cdot 0 & 0 + \frac{225}{712} \cdot 0 & 0 + \frac{225}{712} \cdot 0 & 0 + \frac{225}{712} \cdot 0 & 1 + \frac{225}{712} \cdot 0 & - \frac{225}{712} + \frac{225}{712} \cdot 1 & 0 + \frac{225}{712} \cdot 0 & 0 + \frac{225}{712} \cdot 0 & 0 + \frac{225}{712} \cdot 0 & \frac{2555}{32} + \frac{225}{712} \cdot \frac{445}{4} & 0 + \frac{225}{712} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.39.2

Simplifiez $R_{5}$ .

Étape 2.40

Perform the row operation $R_{4} = R_{4} + \frac{1}{209} R_{6}$ to make the entry at $4, 6$ a $0$ .

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Étape 2.40.1

Perform the row operation $R_{4} = R_{4} + \frac{1}{209} R_{6}$ to make the entry at $4, 6$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & - \frac{15}{56} & 0 & 0 & 0 & 100 & 0 \\ 0 + \frac{1}{209} \cdot 0 & 0 + \frac{1}{209} \cdot 0 & 0 + \frac{1}{209} \cdot 0 & 1 + \frac{1}{209} \cdot 0 & - \frac{60}{209} + \frac{1}{209} \cdot 0 & - \frac{1}{209} + \frac{1}{209} \cdot 1 & 0 + \frac{1}{209} \cdot 0 & 0 + \frac{1}{209} \cdot 0 & 0 + \frac{1}{209} \cdot 0 & \frac{16240}{209} + \frac{1}{209} \cdot \frac{445}{4} & 0 + \frac{1}{209} \cdot 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.40.2

Simplifiez $R_{4}$ .

Étape 2.41

Perform the row operation $R_{3} = R_{3} + \frac{15}{56} R_{6}$ to make the entry at $3, 6$ a $0$ .

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Étape 2.41.1

Perform the row operation $R_{3} = R_{3} + \frac{15}{56} R_{6}$ to make the entry at $3, 6$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 + \frac{15}{56} \cdot 0 & 0 + \frac{15}{56} \cdot 0 & 1 + \frac{15}{56} \cdot 0 & - \frac{1}{56} + \frac{15}{56} \cdot 0 & - \frac{1}{14} + \frac{15}{56} \cdot 0 & - \frac{15}{56} + \frac{15}{56} \cdot 1 & 0 + \frac{15}{56} \cdot 0 & 0 + \frac{15}{56} \cdot 0 & 0 + \frac{15}{56} \cdot 0 & 100 + \frac{15}{56} \cdot \frac{445}{4} & 0 + \frac{15}{56} \cdot 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & 0 & 0 & 0 & 0 & \frac{65405}{836} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.41.2

Simplifiez $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & 0 & 0 & 0 & 0 & \frac{29075}{224} & 0 \\ 0 & 0 & 0 & 1 & - \frac{60}{209} & 0 & 0 & 0 & 0 & \frac{65405}{836} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.42

Perform the row operation $R_{4} = R_{4} + \frac{60}{209} R_{5}$ to make the entry at $4, 5$ a $0$ .

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Étape 2.42.1

Perform the row operation $R_{4} = R_{4} + \frac{60}{209} R_{5}$ to make the entry at $4, 5$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & 0 & 0 & 0 & 0 & \frac{29075}{224} & 0 \\ 0 + \frac{60}{209} \cdot 0 & 0 + \frac{60}{209} \cdot 0 & 0 + \frac{60}{209} \cdot 0 & 1 + \frac{60}{209} \cdot 0 & - \frac{60}{209} + \frac{60}{209} \cdot 1 & 0 + \frac{60}{209} \cdot 0 & 0 + \frac{60}{209} \cdot 0 & 0 + \frac{60}{209} \cdot 0 & 0 + \frac{60}{209} \cdot 0 & \frac{65405}{836} + \frac{60}{209} \cdot 115 & 0 + \frac{60}{209} \cdot 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.42.2

Simplifiez $R_{4}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & - \frac{1}{14} & 0 & 0 & 0 & 0 & \frac{29075}{224} & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.43

Perform the row operation $R_{3} = R_{3} + \frac{1}{14} R_{5}$ to make the entry at $3, 5$ a $0$ .

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Étape 2.43.1

Perform the row operation $R_{3} = R_{3} + \frac{1}{14} R_{5}$ to make the entry at $3, 5$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 + \frac{1}{14} \cdot 0 & 0 + \frac{1}{14} \cdot 0 & 1 + \frac{1}{14} \cdot 0 & - \frac{1}{56} + \frac{1}{14} \cdot 0 & - \frac{1}{14} + \frac{1}{14} \cdot 1 & 0 + \frac{1}{14} \cdot 0 & 0 + \frac{1}{14} \cdot 0 & 0 + \frac{1}{14} \cdot 0 & 0 + \frac{1}{14} \cdot 0 & \frac{29075}{224} + \frac{1}{14} \cdot 115 & 0 + \frac{1}{14} \cdot 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.43.2

Simplifiez $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & - \frac{4}{15} & 0 & 0 & 0 & 0 & \frac{220}{3} & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & 0 & 0 & 0 & 0 & 0 & \frac{30915}{224} & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.44

Perform the row operation $R_{2} = R_{2} + \frac{4}{15} R_{5}$ to make the entry at $2, 5$ a $0$ .

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Étape 2.44.1

Perform the row operation $R_{2} = R_{2} + \frac{4}{15} R_{5}$ to make the entry at $2, 5$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 + \frac{4}{15} \cdot 0 & 1 + \frac{4}{15} \cdot 0 & - \frac{4}{15} + \frac{4}{15} \cdot 0 & - \frac{1}{15} + \frac{4}{15} \cdot 0 & - \frac{4}{15} + \frac{4}{15} \cdot 1 & 0 + \frac{4}{15} \cdot 0 & 0 + \frac{4}{15} \cdot 0 & 0 + \frac{4}{15} \cdot 0 & 0 + \frac{4}{15} \cdot 0 & \frac{220}{3} + \frac{4}{15} \cdot 115 & 0 + \frac{4}{15} \cdot 0 \\ 0 & 0 & 1 & - \frac{1}{56} & 0 & 0 & 0 & 0 & 0 & \frac{30915}{224} & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.44.2

Simplifiez $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & 0 & 0 & 0 & 0 & 0 & 104 & 0 \\ 0 & 0 & 1 & - \frac{1}{56} & 0 & 0 & 0 & 0 & 0 & \frac{30915}{224} & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.45

Perform the row operation $R_{3} = R_{3} + \frac{1}{56} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Étape 2.45.1

Perform the row operation $R_{3} = R_{3} + \frac{1}{56} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & 0 & 0 & 0 & 0 & 0 & 104 & 0 \\ 0 + \frac{1}{56} \cdot 0 & 0 + \frac{1}{56} \cdot 0 & 1 + \frac{1}{56} \cdot 0 & - \frac{1}{56} + \frac{1}{56} \cdot 1 & 0 + \frac{1}{56} \cdot 0 & 0 + \frac{1}{56} \cdot 0 & 0 + \frac{1}{56} \cdot 0 & 0 + \frac{1}{56} \cdot 0 & 0 + \frac{1}{56} \cdot 0 & \frac{30915}{224} + \frac{1}{56} \cdot \frac{445}{4} & 0 + \frac{1}{56} \cdot 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.45.2

Simplifiez $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & 0 & 0 & 0 & 0 & 0 & 104 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 140 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.46

Perform the row operation $R_{2} = R_{2} + \frac{1}{15} R_{4}$ to make the entry at $2, 4$ a $0$ .

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Étape 2.46.1

Perform the row operation $R_{2} = R_{2} + \frac{1}{15} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 + \frac{1}{15} \cdot 0 & 1 + \frac{1}{15} \cdot 0 & - \frac{4}{15} + \frac{1}{15} \cdot 0 & - \frac{1}{15} + \frac{1}{15} \cdot 1 & 0 + \frac{1}{15} \cdot 0 & 0 + \frac{1}{15} \cdot 0 & 0 + \frac{1}{15} \cdot 0 & 0 + \frac{1}{15} \cdot 0 & 0 + \frac{1}{15} \cdot 0 & 104 + \frac{1}{15} \cdot \frac{445}{4} & 0 + \frac{1}{15} \cdot 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 140 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.46.2

Simplifiez $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 75 & 0 \\ 0 & 1 & - \frac{4}{15} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1337}{12} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 140 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.47

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

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Étape 2.47.1

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

$[\begin{array}{c} 1 + \frac{1}{4} \cdot 0 & - \frac{1}{4} + \frac{1}{4} \cdot 0 & 0 + \frac{1}{4} \cdot 0 & - \frac{1}{4} + \frac{1}{4} \cdot 1 & 0 + \frac{1}{4} \cdot 0 & 0 + \frac{1}{4} \cdot 0 & 0 + \frac{1}{4} \cdot 0 & 0 + \frac{1}{4} \cdot 0 & 0 + \frac{1}{4} \cdot 0 & 75 + \frac{1}{4} \cdot \frac{445}{4} & 0 + \frac{1}{4} \cdot 0 \\ 0 & 1 & - \frac{4}{15} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1337}{12} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 140 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.47.2

Simplifiez $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1645}{16} & 0 \\ 0 & 1 & - \frac{4}{15} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1337}{12} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 140 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.48

Perform the row operation $R_{2} = R_{2} + \frac{4}{15} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Étape 2.48.1

Perform the row operation $R_{2} = R_{2} + \frac{4}{15} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1645}{16} & 0 \\ 0 + \frac{4}{15} \cdot 0 & 1 + \frac{4}{15} \cdot 0 & - \frac{4}{15} + \frac{4}{15} \cdot 1 & 0 + \frac{4}{15} \cdot 0 & 0 + \frac{4}{15} \cdot 0 & 0 + \frac{4}{15} \cdot 0 & 0 + \frac{4}{15} \cdot 0 & 0 + \frac{4}{15} \cdot 0 & 0 + \frac{4}{15} \cdot 0 & \frac{1337}{12} + \frac{4}{15} \cdot 140 & 0 + \frac{4}{15} \cdot 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 140 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.48.2

Simplifiez $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1645}{16} & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{595}{4} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 140 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.49

Perform the row operation $R_{1} = R_{1} + \frac{1}{4} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Étape 2.49.1

Perform the row operation $R_{1} = R_{1} + \frac{1}{4} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{1}{4} \cdot 0 & - \frac{1}{4} + \frac{1}{4} \cdot 1 & 0 + \frac{1}{4} \cdot 0 & 0 + \frac{1}{4} \cdot 0 & 0 + \frac{1}{4} \cdot 0 & 0 + \frac{1}{4} \cdot 0 & 0 + \frac{1}{4} \cdot 0 & 0 + \frac{1}{4} \cdot 0 & 0 + \frac{1}{4} \cdot 0 & \frac{1645}{16} + \frac{1}{4} \cdot \frac{595}{4} & 0 + \frac{1}{4} \cdot 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{595}{4} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 140 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 2.49.2

Simplifiez $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 140 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{595}{4} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 140 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 115 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \frac{445}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 90 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{355}{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 90 & 0 \end{array}]$

Étape 3

Use the result matrix to declare the final solution to the system of equations.

$x_{1} + 140 x_{10} = 0$

$x_{2} + \frac{595}{4} x_{10} = 0$

$x_{3} + 140 x_{10} = 0$

$x_{4} + \frac{445}{4} x_{10} = 0$

$x_{5} + 115 x_{10} = 0$

$x_{6} + \frac{445}{4} x_{10} = 0$

$x_{7} + 90 x_{10} = 0$

$x_{8} + \frac{355}{4} x_{10} = 0$

$x_{9} + 90 x_{10} = 0$

Étape 4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \\ x_{6} \\ x_{7} \\ x_{8} \\ x_{9} \\ x_{10} \end{array}] = [\begin{array}{c} - 140 x_{10} \\ - \frac{595 x_{10}}{4} \\ - 140 x_{10} \\ - \frac{445 x_{10}}{4} \\ - 115 x_{10} \\ - \frac{445 x_{10}}{4} \\ - 90 x_{10} \\ - \frac{355 x_{10}}{4} \\ - 90 x_{10} \\ x_{10} \end{array}]$

Étape 5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \\ x_{6} \\ x_{7} \\ x_{8} \\ x_{9} \\ x_{10} \end{array}] = x_{10} [\begin{array}{c} - 140 \\ - \frac{595}{4} \\ - 140 \\ - \frac{445}{4} \\ - 115 \\ - \frac{445}{4} \\ - 90 \\ - \frac{355}{4} \\ - 90 \\ 1 \end{array}]$

Étape 6

Write as a solution set.

${x_{10} [\begin{array}{c} - 140 \\ - \frac{595}{4} \\ - 140 \\ - \frac{445}{4} \\ - 115 \\ - \frac{445}{4} \\ - 90 \\ - \frac{355}{4} \\ - 90 \\ 1 \end{array}] | x_{10} \in R}$