الجبر الخطي الأمثلة

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 1 & - 1 & 1 & - 1 & 1 & 0.5 \\ 1 & 1 & 1 & 1 & 1 & 2 \\ 1 & 2 & 4 & 8 & 16 & 4 \\ 1 & 3 & 9 & 27 & 81 & 8 \end{array}]$

خطوة 1

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

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خطوة 1.1

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

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 1 - 1 & - 1 - 0 & 1 - 0 & - 1 - 0 & 1 - 0 & 0.5 - 1 \\ 1 & 1 & 1 & 1 & 1 & 2 \\ 1 & 2 & 4 & 8 & 16 & 4 \\ 1 & 3 & 9 & 27 & 81 & 8 \end{array}]$

خطوة 1.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & - 1 & 1 & - 1 & 1 & - 0.5 \\ 1 & 1 & 1 & 1 & 1 & 2 \\ 1 & 2 & 4 & 8 & 16 & 4 \\ 1 & 3 & 9 & 27 & 81 & 8 \end{array}]$

خطوة 2

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

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خطوة 2.1

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

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & - 1 & 1 & - 1 & 1 & - 0.5 \\ 1 - 1 & 1 - 0 & 1 - 0 & 1 - 0 & 1 - 0 & 2 - 1 \\ 1 & 2 & 4 & 8 & 16 & 4 \\ 1 & 3 & 9 & 27 & 81 & 8 \end{array}]$

خطوة 2.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & - 1 & 1 & - 1 & 1 & - 0.5 \\ 0 & 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 4 & 8 & 16 & 4 \\ 1 & 3 & 9 & 27 & 81 & 8 \end{array}]$

خطوة 3

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

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خطوة 3.1

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

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & - 1 & 1 & - 1 & 1 & - 0.5 \\ 0 & 1 & 1 & 1 & 1 & 1 \\ 1 - 1 & 2 - 0 & 4 - 0 & 8 - 0 & 16 - 0 & 4 - 1 \\ 1 & 3 & 9 & 27 & 81 & 8 \end{array}]$

خطوة 3.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & - 1 & 1 & - 1 & 1 & - 0.5 \\ 0 & 1 & 1 & 1 & 1 & 1 \\ 0 & 2 & 4 & 8 & 16 & 3 \\ 1 & 3 & 9 & 27 & 81 & 8 \end{array}]$

خطوة 4

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

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خطوة 4.1

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

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & - 1 & 1 & - 1 & 1 & - 0.5 \\ 0 & 1 & 1 & 1 & 1 & 1 \\ 0 & 2 & 4 & 8 & 16 & 3 \\ 1 - 1 & 3 - 0 & 9 - 0 & 27 - 0 & 81 - 0 & 8 - 1 \end{array}]$

خطوة 4.2

بسّط $R_{5}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & - 1 & 1 & - 1 & 1 & - 0.5 \\ 0 & 1 & 1 & 1 & 1 & 1 \\ 0 & 2 & 4 & 8 & 16 & 3 \\ 0 & 3 & 9 & 27 & 81 & 7 \end{array}]$

خطوة 5

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

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خطوة 5.1

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

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ - 0 & - - 1 & - 1 \cdot 1 & - - 1 & - 1 \cdot 1 & - - 0.5 \\ 0 & 1 & 1 & 1 & 1 & 1 \\ 0 & 2 & 4 & 8 & 16 & 3 \\ 0 & 3 & 9 & 27 & 81 & 7 \end{array}]$

خطوة 5.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 1 & 1 & 1 & 1 & 1 \\ 0 & 2 & 4 & 8 & 16 & 3 \\ 0 & 3 & 9 & 27 & 81 & 7 \end{array}]$

خطوة 6

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

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خطوة 6.1

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

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 - 0 & 1 - 1 & 1 + 1 & 1 - 1 & 1 + 1 & 1 - 0.5 \\ 0 & 2 & 4 & 8 & 16 & 3 \\ 0 & 3 & 9 & 27 & 81 & 7 \end{array}]$

خطوة 6.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 2 & 0 & 2 & 0.5 \\ 0 & 2 & 4 & 8 & 16 & 3 \\ 0 & 3 & 9 & 27 & 81 & 7 \end{array}]$

خطوة 7

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

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خطوة 7.1

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

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 2 & 0 & 2 & 0.5 \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & 4 - 2 \cdot - 1 & 8 - 2 \cdot 1 & 16 - 2 \cdot - 1 & 3 - 2 \cdot 0.5 \\ 0 & 3 & 9 & 27 & 81 & 7 \end{array}]$

خطوة 7.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 2 & 0 & 2 & 0.5 \\ 0 & 0 & 6 & 6 & 18 & 2 \\ 0 & 3 & 9 & 27 & 81 & 7 \end{array}]$

خطوة 8

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

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خطوة 8.1

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

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 2 & 0 & 2 & 0.5 \\ 0 & 0 & 6 & 6 & 18 & 2 \\ 0 - 3 \cdot 0 & 3 - 3 \cdot 1 & 9 - 3 \cdot - 1 & 27 - 3 \cdot 1 & 81 - 3 \cdot - 1 & 7 - 3 \cdot 0.5 \end{array}]$

خطوة 8.2

بسّط $R_{5}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 2 & 0 & 2 & 0.5 \\ 0 & 0 & 6 & 6 & 18 & 2 \\ 0 & 0 & 12 & 24 & 84 & 5.5 \end{array}]$

خطوة 9

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

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خطوة 9.1

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

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ \frac{0}{2} & \frac{0}{2} & \frac{2}{2} & \frac{0}{2} & \frac{2}{2} & \frac{0.5}{2} \\ 0 & 0 & 6 & 6 & 18 & 2 \\ 0 & 0 & 12 & 24 & 84 & 5.5 \end{array}]$

خطوة 9.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 1 & 0 & 1 & 0.25 \\ 0 & 0 & 6 & 6 & 18 & 2 \\ 0 & 0 & 12 & 24 & 84 & 5.5 \end{array}]$

خطوة 10

Perform the row operation $R_{4} = R_{4} - 6 R_{3}$ to make the entry at $4, 3$ a $0$ .

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خطوة 10.1

Perform the row operation $R_{4} = R_{4} - 6 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 1 & 0 & 1 & 0.25 \\ 0 - 6 \cdot 0 & 0 - 6 \cdot 0 & 6 - 6 \cdot 1 & 6 - 6 \cdot 0 & 18 - 6 \cdot 1 & 2 - 6 \cdot 0.25 \\ 0 & 0 & 12 & 24 & 84 & 5.5 \end{array}]$

خطوة 10.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 1 & 0 & 1 & 0.25 \\ 0 & 0 & 0 & 6 & 12 & 0.5 \\ 0 & 0 & 12 & 24 & 84 & 5.5 \end{array}]$

خطوة 11

Perform the row operation $R_{5} = R_{5} - 12 R_{3}$ to make the entry at $5, 3$ a $0$ .

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خطوة 11.1

Perform the row operation $R_{5} = R_{5} - 12 R_{3}$ to make the entry at $5, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 1 & 0 & 1 & 0.25 \\ 0 & 0 & 0 & 6 & 12 & 0.5 \\ 0 - 12 \cdot 0 & 0 - 12 \cdot 0 & 12 - 12 \cdot 1 & 24 - 12 \cdot 0 & 84 - 12 \cdot 1 & 5.5 - 12 \cdot 0.25 \end{array}]$

خطوة 11.2

بسّط $R_{5}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 1 & 0 & 1 & 0.25 \\ 0 & 0 & 0 & 6 & 12 & 0.5 \\ 0 & 0 & 0 & 24 & 72 & 2.5 \end{array}]$

خطوة 12

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

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خطوة 12.1

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

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 1 & 0 & 1 & 0.25 \\ \frac{0}{6} & \frac{0}{6} & \frac{0}{6} & \frac{6}{6} & \frac{12}{6} & \frac{0.5}{6} \\ 0 & 0 & 0 & 24 & 72 & 2.5 \end{array}]$

خطوة 12.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 1 & 0 & 1 & 0.25 \\ 0 & 0 & 0 & 1 & 2 & 0.08\overline{3} \\ 0 & 0 & 0 & 24 & 72 & 2.5 \end{array}]$

خطوة 13

Perform the row operation $R_{5} = R_{5} - 24 R_{4}$ to make the entry at $5, 4$ a $0$ .

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خطوة 13.1

Perform the row operation $R_{5} = R_{5} - 24 R_{4}$ to make the entry at $5, 4$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 1 & 0 & 1 & 0.25 \\ 0 & 0 & 0 & 1 & 2 & 0.08\overline{3} \\ 0 - 24 \cdot 0 & 0 - 24 \cdot 0 & 0 - 24 \cdot 0 & 24 - 24 \cdot 1 & 72 - 24 \cdot 2 & 2.5 - 24 \cdot 0.08\overline{3} \end{array}]$

خطوة 13.2

بسّط $R_{5}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 1 & 0 & 1 & 0.25 \\ 0 & 0 & 0 & 1 & 2 & 0.08\overline{3} \\ 0 & 0 & 0 & 0 & 24 & 0.5 \end{array}]$

خطوة 14

Multiply each element of $R_{5}$ by $\frac{1}{24}$ to make the entry at $5, 5$ a $1$ .

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خطوة 14.1

Multiply each element of $R_{5}$ by $\frac{1}{24}$ to make the entry at $5, 5$ a $1$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 1 & 0 & 1 & 0.25 \\ 0 & 0 & 0 & 1 & 2 & 0.08\overline{3} \\ \frac{0}{24} & \frac{0}{24} & \frac{0}{24} & \frac{0}{24} & \frac{24}{24} & \frac{0.5}{24} \end{array}]$

خطوة 14.2

بسّط $R_{5}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 1 & 0 & 1 & 0.25 \\ 0 & 0 & 0 & 1 & 2 & 0.08\overline{3} \\ 0 & 0 & 0 & 0 & 1 & 0.0208\overline{3} \end{array}]$

خطوة 15

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

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خطوة 15.1

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

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 1 & 0 & 1 & 0.25 \\ 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0.08\overline{3} - 2 \cdot 0.0208\overline{3} \\ 0 & 0 & 0 & 0 & 1 & 0.0208\overline{3} \end{array}]$

خطوة 15.2

بسّط $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 1 & 0 & 1 & 0.25 \\ 0 & 0 & 0 & 1 & 0 & 0.041\overline{6} \\ 0 & 0 & 0 & 0 & 1 & 0.0208\overline{3} \end{array}]$

خطوة 16

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

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خطوة 16.1

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

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 - 0 & 0 - 0 & 1 - 0 & 0 - 0 & 1 - 1 & 0.25 - 0.0208\overline{3} \\ 0 & 0 & 0 & 1 & 0 & 0.041\overline{6} \\ 0 & 0 & 0 & 0 & 1 & 0.0208\overline{3} \end{array}]$

خطوة 16.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 0.5 \\ 0 & 0 & 1 & 0 & 0 & 0.2291\overline{6} \\ 0 & 0 & 0 & 1 & 0 & 0.041\overline{6} \\ 0 & 0 & 0 & 0 & 1 & 0.0208\overline{3} \end{array}]$

خطوة 17

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

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خطوة 17.1

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

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 + 0 & 1 + 0 & - 1 + 0 & 1 + 0 & - 1 + 1 \cdot 1 & 0.5 + 1 \cdot 0.0208\overline{3} \\ 0 & 0 & 1 & 0 & 0 & 0.2291\overline{6} \\ 0 & 0 & 0 & 1 & 0 & 0.041\overline{6} \\ 0 & 0 & 0 & 0 & 1 & 0.0208\overline{3} \end{array}]$

خطوة 17.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & 0 & 0.5208\overline{3} \\ 0 & 0 & 1 & 0 & 0 & 0.2291\overline{6} \\ 0 & 0 & 0 & 1 & 0 & 0.041\overline{6} \\ 0 & 0 & 0 & 0 & 1 & 0.0208\overline{3} \end{array}]$

خطوة 18

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

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خطوة 18.1

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

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 - 0 & 1 - 0 & - 1 - 0 & 1 - 1 & 0 - 0 & 0.5208\overline{3} - 0.041\overline{6} \\ 0 & 0 & 1 & 0 & 0 & 0.2291\overline{6} \\ 0 & 0 & 0 & 1 & 0 & 0.041\overline{6} \\ 0 & 0 & 0 & 0 & 1 & 0.0208\overline{3} \end{array}]$

خطوة 18.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 0 & 0 & 0.4791\overline{6} \\ 0 & 0 & 1 & 0 & 0 & 0.2291\overline{6} \\ 0 & 0 & 0 & 1 & 0 & 0.041\overline{6} \\ 0 & 0 & 0 & 0 & 1 & 0.0208\overline{3} \end{array}]$

خطوة 19

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

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خطوة 19.1

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

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 + 0 & 1 + 0 & - 1 + 1 \cdot 1 & 0 + 0 & 0 + 0 & 0.4791\overline{6} + 1 \cdot 0.2291\overline{6} \\ 0 & 0 & 1 & 0 & 0 & 0.2291\overline{6} \\ 0 & 0 & 0 & 1 & 0 & 0.041\overline{6} \\ 0 & 0 & 0 & 0 & 1 & 0.0208\overline{3} \end{array}]$

خطوة 19.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0.708\overline{3} \\ 0 & 0 & 1 & 0 & 0 & 0.2291\overline{6} \\ 0 & 0 & 0 & 1 & 0 & 0.041\overline{6} \\ 0 & 0 & 0 & 0 & 1 & 0.0208\overline{3} \end{array}]$