الجبر الخطي الأمثلة

$[\begin{array}{c} 3 & 2 & 1 & 1 & 0 & 0 & 0 & 40 \\ 1 & 6 & 4 & 0 & 1 & 0 & 0 & 60 \\ 5 & 1 & 4 & 0 & 0 & 1 & 0 & 90 \\ - 6 & - 5 & - 9 & 0 & 0 & 0 & 1 & 0 \end{array}]$

خطوة 1

أوجِد الصيغة الدرجية المختزلة صفيًا.

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خطوة 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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خطوة 1.1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & \frac{0}{3} & \frac{0}{3} & \frac{0}{3} & \frac{40}{3} \\ 1 & 6 & 4 & 0 & 1 & 0 & 0 & 60 \\ 5 & 1 & 4 & 0 & 0 & 1 & 0 & 90 \\ - 6 & - 5 & - 9 & 0 & 0 & 0 & 1 & 0 \end{array}]$

خطوة 1.1.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{40}{3} \\ 1 & 6 & 4 & 0 & 1 & 0 & 0 & 60 \\ 5 & 1 & 4 & 0 & 0 & 1 & 0 & 90 \\ - 6 & - 5 & - 9 & 0 & 0 & 0 & 1 & 0 \end{array}]$

خطوة 1.2

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

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خطوة 1.2.1

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

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{40}{3} \\ 1 - 1 & 6 - \frac{2}{3} & 4 - \frac{1}{3} & 0 - \frac{1}{3} & 1 - 0 & 0 - 0 & 0 - 0 & 60 - \frac{40}{3} \\ 5 & 1 & 4 & 0 & 0 & 1 & 0 & 90 \\ - 6 & - 5 & - 9 & 0 & 0 & 0 & 1 & 0 \end{array}]$

خطوة 1.2.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{40}{3} \\ 0 & \frac{16}{3} & \frac{11}{3} & - \frac{1}{3} & 1 & 0 & 0 & \frac{140}{3} \\ 5 & 1 & 4 & 0 & 0 & 1 & 0 & 90 \\ - 6 & - 5 & - 9 & 0 & 0 & 0 & 1 & 0 \end{array}]$

خطوة 1.3

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

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خطوة 1.3.1

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

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{40}{3} \\ 0 & \frac{16}{3} & \frac{11}{3} & - \frac{1}{3} & 1 & 0 & 0 & \frac{140}{3} \\ 5 - 5 \cdot 1 & 1 - 5 (\frac{2}{3}) & 4 - 5 (\frac{1}{3}) & 0 - 5 (\frac{1}{3}) & 0 - 5 \cdot 0 & 1 - 5 \cdot 0 & 0 - 5 \cdot 0 & 90 - 5 (\frac{40}{3}) \\ - 6 & - 5 & - 9 & 0 & 0 & 0 & 1 & 0 \end{array}]$

خطوة 1.3.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{40}{3} \\ 0 & \frac{16}{3} & \frac{11}{3} & - \frac{1}{3} & 1 & 0 & 0 & \frac{140}{3} \\ 0 & - \frac{7}{3} & \frac{7}{3} & - \frac{5}{3} & 0 & 1 & 0 & \frac{70}{3} \\ - 6 & - 5 & - 9 & 0 & 0 & 0 & 1 & 0 \end{array}]$

خطوة 1.4

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

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خطوة 1.4.1

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

خطوة 1.4.2

بسّط $R_{4}$ .

خطوة 1.5

Multiply each element of $R_{2}$ by $\frac{3}{16}$ to make the entry at $2, 2$ a $1$ .

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خطوة 1.5.1

Multiply each element of $R_{2}$ by $\frac{3}{16}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{40}{3} \\ \frac{3}{16} \cdot 0 & \frac{3}{16} \cdot \frac{16}{3} & \frac{3}{16} \cdot \frac{11}{3} & \frac{3}{16} (- \frac{1}{3}) & \frac{3}{16} \cdot 1 & \frac{3}{16} \cdot 0 & \frac{3}{16} \cdot 0 & \frac{3}{16} \cdot \frac{140}{3} \\ 0 & - \frac{7}{3} & \frac{7}{3} & - \frac{5}{3} & 0 & 1 & 0 & \frac{70}{3} \\ 0 & - 1 & - 7 & 2 & 0 & 0 & 1 & 80 \end{array}]$

خطوة 1.5.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{40}{3} \\ 0 & 1 & \frac{11}{16} & - \frac{1}{16} & \frac{3}{16} & 0 & 0 & \frac{35}{4} \\ 0 & - \frac{7}{3} & \frac{7}{3} & - \frac{5}{3} & 0 & 1 & 0 & \frac{70}{3} \\ 0 & - 1 & - 7 & 2 & 0 & 0 & 1 & 80 \end{array}]$

خطوة 1.6

Perform the row operation $R_{3} = R_{3} + \frac{7}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

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خطوة 1.6.1

Perform the row operation $R_{3} = R_{3} + \frac{7}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{40}{3} \\ 0 & 1 & \frac{11}{16} & - \frac{1}{16} & \frac{3}{16} & 0 & 0 & \frac{35}{4} \\ 0 + \frac{7}{3} \cdot 0 & - \frac{7}{3} + \frac{7}{3} \cdot 1 & \frac{7}{3} + \frac{7}{3} \cdot \frac{11}{16} & - \frac{5}{3} + \frac{7}{3} (- \frac{1}{16}) & 0 + \frac{7}{3} \cdot \frac{3}{16} & 1 + \frac{7}{3} \cdot 0 & 0 + \frac{7}{3} \cdot 0 & \frac{70}{3} + \frac{7}{3} \cdot \frac{35}{4} \\ 0 & - 1 & - 7 & 2 & 0 & 0 & 1 & 80 \end{array}]$

خطوة 1.6.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{40}{3} \\ 0 & 1 & \frac{11}{16} & - \frac{1}{16} & \frac{3}{16} & 0 & 0 & \frac{35}{4} \\ 0 & 0 & \frac{63}{16} & - \frac{29}{16} & \frac{7}{16} & 1 & 0 & \frac{175}{4} \\ 0 & - 1 & - 7 & 2 & 0 & 0 & 1 & 80 \end{array}]$

خطوة 1.7

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

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خطوة 1.7.1

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

خطوة 1.7.2

بسّط $R_{4}$ .

خطوة 1.8

Multiply each element of $R_{3}$ by $\frac{16}{63}$ to make the entry at $3, 3$ a $1$ .

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خطوة 1.8.1

Multiply each element of $R_{3}$ by $\frac{16}{63}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{40}{3} \\ 0 & 1 & \frac{11}{16} & - \frac{1}{16} & \frac{3}{16} & 0 & 0 & \frac{35}{4} \\ \frac{16}{63} \cdot 0 & \frac{16}{63} \cdot 0 & \frac{16}{63} \cdot \frac{63}{16} & \frac{16}{63} (- \frac{29}{16}) & \frac{16}{63} \cdot \frac{7}{16} & \frac{16}{63} \cdot 1 & \frac{16}{63} \cdot 0 & \frac{16}{63} \cdot \frac{175}{4} \\ 0 & 0 & - \frac{101}{16} & \frac{31}{16} & \frac{3}{16} & 0 & 1 & \frac{355}{4} \end{array}]$

خطوة 1.8.2

بسّط $R_{3}$ .

خطوة 1.9

Perform the row operation $R_{4} = R_{4} + \frac{101}{16} R_{3}$ to make the entry at $4, 3$ a $0$ .

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خطوة 1.9.1

Perform the row operation $R_{4} = R_{4} + \frac{101}{16} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{40}{3} \\ 0 & 1 & \frac{11}{16} & - \frac{1}{16} & \frac{3}{16} & 0 & 0 & \frac{35}{4} \\ 0 & 0 & 1 & - \frac{29}{63} & \frac{1}{9} & \frac{16}{63} & 0 & \frac{100}{9} \\ 0 + \frac{101}{16} \cdot 0 & 0 + \frac{101}{16} \cdot 0 & - \frac{101}{16} + \frac{101}{16} \cdot 1 & \frac{31}{16} + \frac{101}{16} (- \frac{29}{63}) & \frac{3}{16} + \frac{101}{16} \cdot \frac{1}{9} & 0 + \frac{101}{16} \cdot \frac{16}{63} & 1 + \frac{101}{16} \cdot 0 & \frac{355}{4} + \frac{101}{16} \cdot \frac{100}{9} \end{array}]$

خطوة 1.9.2

بسّط $R_{4}$ .

خطوة 1.10

Multiply each element of $R_{4}$ by $- \frac{63}{61}$ to make the entry at $4, 4$ a $1$ .

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خطوة 1.10.1

Multiply each element of $R_{4}$ by $- \frac{63}{61}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{40}{3} \\ 0 & 1 & \frac{11}{16} & - \frac{1}{16} & \frac{3}{16} & 0 & 0 & \frac{35}{4} \\ 0 & 0 & 1 & - \frac{29}{63} & \frac{1}{9} & \frac{16}{63} & 0 & \frac{100}{9} \\ - \frac{63}{61} \cdot 0 & - \frac{63}{61} \cdot 0 & - \frac{63}{61} \cdot 0 & - \frac{63}{61} (- \frac{61}{63}) & - \frac{63}{61} \cdot \frac{8}{9} & - \frac{63}{61} \cdot \frac{101}{63} & - \frac{63}{61} \cdot 1 & - \frac{63}{61} \cdot \frac{1430}{9} \end{array}]$

خطوة 1.10.2

بسّط $R_{4}$ .

خطوة 1.11

Perform the row operation $R_{3} = R_{3} + \frac{29}{63} R_{4}$ to make the entry at $3, 4$ a $0$ .

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خطوة 1.11.1

Perform the row operation $R_{3} = R_{3} + \frac{29}{63} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{40}{3} \\ 0 & 1 & \frac{11}{16} & - \frac{1}{16} & \frac{3}{16} & 0 & 0 & \frac{35}{4} \\ 0 + \frac{29}{63} \cdot 0 & 0 + \frac{29}{63} \cdot 0 & 1 + \frac{29}{63} \cdot 0 & - \frac{29}{63} + \frac{29}{63} \cdot 1 & \frac{1}{9} + \frac{29}{63} (- \frac{56}{61}) & \frac{16}{63} + \frac{29}{63} (- \frac{101}{61}) & 0 + \frac{29}{63} (- \frac{63}{61}) & \frac{100}{9} + \frac{29}{63} (- \frac{10010}{61}) \\ 0 & 0 & 0 & 1 & - \frac{56}{61} & - \frac{101}{61} & - \frac{63}{61} & - \frac{10010}{61} \end{array}]$

خطوة 1.11.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{40}{3} \\ 0 & 1 & \frac{11}{16} & - \frac{1}{16} & \frac{3}{16} & 0 & 0 & \frac{35}{4} \\ 0 & 0 & 1 & 0 & - \frac{19}{61} & - \frac{31}{61} & - \frac{29}{61} & - \frac{3930}{61} \\ 0 & 0 & 0 & 1 & - \frac{56}{61} & - \frac{101}{61} & - \frac{63}{61} & - \frac{10010}{61} \end{array}]$

خطوة 1.12

Perform the row operation $R_{2} = R_{2} + \frac{1}{16} R_{4}$ to make the entry at $2, 4$ a $0$ .

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خطوة 1.12.1

Perform the row operation $R_{2} = R_{2} + \frac{1}{16} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{40}{3} \\ 0 + \frac{1}{16} \cdot 0 & 1 + \frac{1}{16} \cdot 0 & \frac{11}{16} + \frac{1}{16} \cdot 0 & - \frac{1}{16} + \frac{1}{16} \cdot 1 & \frac{3}{16} + \frac{1}{16} (- \frac{56}{61}) & 0 + \frac{1}{16} (- \frac{101}{61}) & 0 + \frac{1}{16} (- \frac{63}{61}) & \frac{35}{4} + \frac{1}{16} (- \frac{10010}{61}) \\ 0 & 0 & 1 & 0 & - \frac{19}{61} & - \frac{31}{61} & - \frac{29}{61} & - \frac{3930}{61} \\ 0 & 0 & 0 & 1 & - \frac{56}{61} & - \frac{101}{61} & - \frac{63}{61} & - \frac{10010}{61} \end{array}]$

خطوة 1.12.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & \frac{40}{3} \\ 0 & 1 & \frac{11}{16} & 0 & \frac{127}{976} & - \frac{101}{976} & - \frac{63}{976} & - \frac{735}{488} \\ 0 & 0 & 1 & 0 & - \frac{19}{61} & - \frac{31}{61} & - \frac{29}{61} & - \frac{3930}{61} \\ 0 & 0 & 0 & 1 & - \frac{56}{61} & - \frac{101}{61} & - \frac{63}{61} & - \frac{10010}{61} \end{array}]$

خطوة 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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خطوة 1.13.1

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

$[\begin{array}{c} 1 - \frac{1}{3} \cdot 0 & \frac{2}{3} - \frac{1}{3} \cdot 0 & \frac{1}{3} - \frac{1}{3} \cdot 0 & \frac{1}{3} - \frac{1}{3} \cdot 1 & 0 - \frac{1}{3} (- \frac{56}{61}) & 0 - \frac{1}{3} (- \frac{101}{61}) & 0 - \frac{1}{3} (- \frac{63}{61}) & \frac{40}{3} - \frac{1}{3} (- \frac{10010}{61}) \\ 0 & 1 & \frac{11}{16} & 0 & \frac{127}{976} & - \frac{101}{976} & - \frac{63}{976} & - \frac{735}{488} \\ 0 & 0 & 1 & 0 & - \frac{19}{61} & - \frac{31}{61} & - \frac{29}{61} & - \frac{3930}{61} \\ 0 & 0 & 0 & 1 & - \frac{56}{61} & - \frac{101}{61} & - \frac{63}{61} & - \frac{10010}{61} \end{array}]$

خطوة 1.13.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & 0 & \frac{56}{183} & \frac{101}{183} & \frac{21}{61} & \frac{4150}{61} \\ 0 & 1 & \frac{11}{16} & 0 & \frac{127}{976} & - \frac{101}{976} & - \frac{63}{976} & - \frac{735}{488} \\ 0 & 0 & 1 & 0 & - \frac{19}{61} & - \frac{31}{61} & - \frac{29}{61} & - \frac{3930}{61} \\ 0 & 0 & 0 & 1 & - \frac{56}{61} & - \frac{101}{61} & - \frac{63}{61} & - \frac{10010}{61} \end{array}]$

خطوة 1.14

Perform the row operation $R_{2} = R_{2} - \frac{11}{16} R_{3}$ to make the entry at $2, 3$ a $0$ .

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خطوة 1.14.1

Perform the row operation $R_{2} = R_{2} - \frac{11}{16} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & 0 & \frac{56}{183} & \frac{101}{183} & \frac{21}{61} & \frac{4150}{61} \\ 0 - \frac{11}{16} \cdot 0 & 1 - \frac{11}{16} \cdot 0 & \frac{11}{16} - \frac{11}{16} \cdot 1 & 0 - \frac{11}{16} \cdot 0 & \frac{127}{976} - \frac{11}{16} (- \frac{19}{61}) & - \frac{101}{976} - \frac{11}{16} (- \frac{31}{61}) & - \frac{63}{976} - \frac{11}{16} (- \frac{29}{61}) & - \frac{735}{488} - \frac{11}{16} (- \frac{3930}{61}) \\ 0 & 0 & 1 & 0 & - \frac{19}{61} & - \frac{31}{61} & - \frac{29}{61} & - \frac{3930}{61} \\ 0 & 0 & 0 & 1 & - \frac{56}{61} & - \frac{101}{61} & - \frac{63}{61} & - \frac{10010}{61} \end{array}]$

خطوة 1.14.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{1}{3} & 0 & \frac{56}{183} & \frac{101}{183} & \frac{21}{61} & \frac{4150}{61} \\ 0 & 1 & 0 & 0 & \frac{21}{61} & \frac{15}{61} & \frac{16}{61} & \frac{2610}{61} \\ 0 & 0 & 1 & 0 & - \frac{19}{61} & - \frac{31}{61} & - \frac{29}{61} & - \frac{3930}{61} \\ 0 & 0 & 0 & 1 & - \frac{56}{61} & - \frac{101}{61} & - \frac{63}{61} & - \frac{10010}{61} \end{array}]$

خطوة 1.15

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

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خطوة 1.15.1

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

$[\begin{array}{c} 1 - \frac{1}{3} \cdot 0 & \frac{2}{3} - \frac{1}{3} \cdot 0 & \frac{1}{3} - \frac{1}{3} \cdot 1 & 0 - \frac{1}{3} \cdot 0 & \frac{56}{183} - \frac{1}{3} (- \frac{19}{61}) & \frac{101}{183} - \frac{1}{3} (- \frac{31}{61}) & \frac{21}{61} - \frac{1}{3} (- \frac{29}{61}) & \frac{4150}{61} - \frac{1}{3} (- \frac{3930}{61}) \\ 0 & 1 & 0 & 0 & \frac{21}{61} & \frac{15}{61} & \frac{16}{61} & \frac{2610}{61} \\ 0 & 0 & 1 & 0 & - \frac{19}{61} & - \frac{31}{61} & - \frac{29}{61} & - \frac{3930}{61} \\ 0 & 0 & 0 & 1 & - \frac{56}{61} & - \frac{101}{61} & - \frac{63}{61} & - \frac{10010}{61} \end{array}]$

خطوة 1.15.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & 0 & 0 & \frac{25}{61} & \frac{44}{61} & \frac{92}{183} & \frac{5460}{61} \\ 0 & 1 & 0 & 0 & \frac{21}{61} & \frac{15}{61} & \frac{16}{61} & \frac{2610}{61} \\ 0 & 0 & 1 & 0 & - \frac{19}{61} & - \frac{31}{61} & - \frac{29}{61} & - \frac{3930}{61} \\ 0 & 0 & 0 & 1 & - \frac{56}{61} & - \frac{101}{61} & - \frac{63}{61} & - \frac{10010}{61} \end{array}]$

خطوة 1.16

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

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خطوة 1.16.1

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

$[\begin{array}{c} 1 - \frac{2}{3} \cdot 0 & \frac{2}{3} - \frac{2}{3} \cdot 1 & 0 - \frac{2}{3} \cdot 0 & 0 - \frac{2}{3} \cdot 0 & \frac{25}{61} - \frac{2}{3} \cdot \frac{21}{61} & \frac{44}{61} - \frac{2}{3} \cdot \frac{15}{61} & \frac{92}{183} - \frac{2}{3} \cdot \frac{16}{61} & \frac{5460}{61} - \frac{2}{3} \cdot \frac{2610}{61} \\ 0 & 1 & 0 & 0 & \frac{21}{61} & \frac{15}{61} & \frac{16}{61} & \frac{2610}{61} \\ 0 & 0 & 1 & 0 & - \frac{19}{61} & - \frac{31}{61} & - \frac{29}{61} & - \frac{3930}{61} \\ 0 & 0 & 0 & 1 & - \frac{56}{61} & - \frac{101}{61} & - \frac{63}{61} & - \frac{10010}{61} \end{array}]$

خطوة 1.16.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & \frac{11}{61} & \frac{34}{61} & \frac{20}{61} & \frac{3720}{61} \\ 0 & 1 & 0 & 0 & \frac{21}{61} & \frac{15}{61} & \frac{16}{61} & \frac{2610}{61} \\ 0 & 0 & 1 & 0 & - \frac{19}{61} & - \frac{31}{61} & - \frac{29}{61} & - \frac{3930}{61} \\ 0 & 0 & 0 & 1 & - \frac{56}{61} & - \frac{101}{61} & - \frac{63}{61} & - \frac{10010}{61} \end{array}]$

خطوة 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 $4$