Linear Algebra Examples

$[\begin{array}{c} 4 & 5 & 9 & 1 & 0 & 0 & 0 & 700 \\ 6 & 1 & 8 & 0 & 1 & 0 & 0 & 300 \\ 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 \\ - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 1

Find the reduced row echelon form.

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Step 1.1

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

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Step 1.1.1

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

$[\begin{array}{c} \frac{4}{4} & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & \frac{0}{4} & \frac{0}{4} & \frac{0}{4} & \frac{700}{4} \\ 6 & 1 & 8 & 0 & 1 & 0 & 0 & 300 \\ 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 \\ - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 1.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 6 & 1 & 8 & 0 & 1 & 0 & 0 & 300 \\ 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 \\ - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 1.2

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

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Step 1.2.1

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

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 6 - 6 \cdot 1 & 1 - 6 (\frac{5}{4}) & 8 - 6 (\frac{9}{4}) & 0 - 6 (\frac{1}{4}) & 1 - 6 \cdot 0 & 0 - 6 \cdot 0 & 0 - 6 \cdot 0 & 300 - 6 \cdot 175 \\ 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 \\ - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 1.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 \\ 8 & 5 & 4 & 0 & 0 & 1 & 0 & 700 \\ - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 1.3

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 1.3.1

Perform the row operation $R_{3} = R_{3} - 8 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 \\ 8 - 8 \cdot 1 & 5 - 8 (\frac{5}{4}) & 4 - 8 (\frac{9}{4}) & 0 - 8 (\frac{1}{4}) & 0 - 8 \cdot 0 & 1 - 8 \cdot 0 & 0 - 8 \cdot 0 & 700 - 8 \cdot 175 \\ - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 1.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 \\ 0 & - 5 & - 14 & - 2 & 0 & 1 & 0 & - 700 \\ - 3 & - 8 & - 1 & 0 & 0 & 0 & 1 & 0 \end{array}]$

Step 1.4

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

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Step 1.4.1

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

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 \\ 0 & - 5 & - 14 & - 2 & 0 & 1 & 0 & - 700 \\ - 3 + 3 \cdot 1 & - 8 + 3 (\frac{5}{4}) & - 1 + 3 (\frac{9}{4}) & 0 + 3 (\frac{1}{4}) & 0 + 3 \cdot 0 & 0 + 3 \cdot 0 & 1 + 3 \cdot 0 & 0 + 3 \cdot 175 \end{array}]$

Step 1.4.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & - \frac{13}{2} & - \frac{11}{2} & - \frac{3}{2} & 1 & 0 & 0 & - 750 \\ 0 & - 5 & - 14 & - 2 & 0 & 1 & 0 & - 700 \\ 0 & - \frac{17}{4} & \frac{23}{4} & \frac{3}{4} & 0 & 0 & 1 & 525 \end{array}]$

Step 1.5

Multiply each element of $R_{2}$ by $- \frac{2}{13}$ to make the entry at $2, 2$ a $1$ .

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Step 1.5.1

Multiply each element of $R_{2}$ by $- \frac{2}{13}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ - \frac{2}{13} \cdot 0 & - \frac{2}{13} (- \frac{13}{2}) & - \frac{2}{13} (- \frac{11}{2}) & - \frac{2}{13} (- \frac{3}{2}) & - \frac{2}{13} \cdot 1 & - \frac{2}{13} \cdot 0 & - \frac{2}{13} \cdot 0 & - \frac{2}{13} \cdot - 750 \\ 0 & - 5 & - 14 & - 2 & 0 & 1 & 0 & - 700 \\ 0 & - \frac{17}{4} & \frac{23}{4} & \frac{3}{4} & 0 & 0 & 1 & 525 \end{array}]$

Step 1.5.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & \frac{3}{13} & - \frac{2}{13} & 0 & 0 & \frac{1500}{13} \\ 0 & - 5 & - 14 & - 2 & 0 & 1 & 0 & - 700 \\ 0 & - \frac{17}{4} & \frac{23}{4} & \frac{3}{4} & 0 & 0 & 1 & 525 \end{array}]$

Step 1.6

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

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Step 1.6.1

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

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & \frac{3}{13} & - \frac{2}{13} & 0 & 0 & \frac{1500}{13} \\ 0 + 5 \cdot 0 & - 5 + 5 \cdot 1 & - 14 + 5 (\frac{11}{13}) & - 2 + 5 (\frac{3}{13}) & 0 + 5 (- \frac{2}{13}) & 1 + 5 \cdot 0 & 0 + 5 \cdot 0 & - 700 + 5 (\frac{1500}{13}) \\ 0 & - \frac{17}{4} & \frac{23}{4} & \frac{3}{4} & 0 & 0 & 1 & 525 \end{array}]$

Step 1.6.2

Simplify $R_{3}$ .

Step 1.7

Perform the row operation $R_{4} = R_{4} + \frac{17}{4} R_{2}$ to make the entry at $4, 2$ a $0$ .

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Step 1.7.1

Perform the row operation $R_{4} = R_{4} + \frac{17}{4} R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & \frac{3}{13} & - \frac{2}{13} & 0 & 0 & \frac{1500}{13} \\ 0 & 0 & - \frac{127}{13} & - \frac{11}{13} & - \frac{10}{13} & 1 & 0 & - \frac{1600}{13} \\ 0 + \frac{17}{4} \cdot 0 & - \frac{17}{4} + \frac{17}{4} \cdot 1 & \frac{23}{4} + \frac{17}{4} \cdot \frac{11}{13} & \frac{3}{4} + \frac{17}{4} \cdot \frac{3}{13} & 0 + \frac{17}{4} (- \frac{2}{13}) & 0 + \frac{17}{4} \cdot 0 & 1 + \frac{17}{4} \cdot 0 & 525 + \frac{17}{4} \cdot \frac{1500}{13} \end{array}]$

Step 1.7.2

Simplify $R_{4}$ .

Step 1.8

Multiply each element of $R_{3}$ by $- \frac{13}{127}$ to make the entry at $3, 3$ a $1$ .

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Step 1.8.1

Multiply each element of $R_{3}$ by $- \frac{13}{127}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & \frac{3}{13} & - \frac{2}{13} & 0 & 0 & \frac{1500}{13} \\ - \frac{13}{127} \cdot 0 & - \frac{13}{127} \cdot 0 & - \frac{13}{127} (- \frac{127}{13}) & - \frac{13}{127} (- \frac{11}{13}) & - \frac{13}{127} (- \frac{10}{13}) & - \frac{13}{127} \cdot 1 & - \frac{13}{127} \cdot 0 & - \frac{13}{127} (- \frac{1600}{13}) \\ 0 & 0 & \frac{243}{26} & \frac{45}{26} & - \frac{17}{26} & 0 & 1 & \frac{13200}{13} \end{array}]$

Step 1.8.2

Simplify $R_{3}$ .

Step 1.9

Perform the row operation $R_{4} = R_{4} - \frac{243}{26} R_{3}$ to make the entry at $4, 3$ a $0$ .

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Step 1.9.1

Perform the row operation $R_{4} = R_{4} - \frac{243}{26} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & \frac{3}{13} & - \frac{2}{13} & 0 & 0 & \frac{1500}{13} \\ 0 & 0 & 1 & \frac{11}{127} & \frac{10}{127} & - \frac{13}{127} & 0 & \frac{1600}{127} \\ 0 - \frac{243}{26} \cdot 0 & 0 - \frac{243}{26} \cdot 0 & \frac{243}{26} - \frac{243}{26} \cdot 1 & \frac{45}{26} - \frac{243}{26} \cdot \frac{11}{127} & - \frac{17}{26} - \frac{243}{26} \cdot \frac{10}{127} & 0 - \frac{243}{26} (- \frac{13}{127}) & 1 - \frac{243}{26} \cdot 0 & \frac{13200}{13} - \frac{243}{26} \cdot \frac{1600}{127} \end{array}]$

Step 1.9.2

Simplify $R_{4}$ .

Step 1.10

Multiply each element of $R_{4}$ by $\frac{127}{117}$ to make the entry at $4, 4$ a $1$ .

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Step 1.10.1

Multiply each element of $R_{4}$ by $\frac{127}{117}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & \frac{3}{13} & - \frac{2}{13} & 0 & 0 & \frac{1500}{13} \\ 0 & 0 & 1 & \frac{11}{127} & \frac{10}{127} & - \frac{13}{127} & 0 & \frac{1600}{127} \\ \frac{127}{117} \cdot 0 & \frac{127}{117} \cdot 0 & \frac{127}{117} \cdot 0 & \frac{127}{117} \cdot \frac{117}{127} & \frac{127}{117} (- \frac{353}{254}) & \frac{127}{117} \cdot \frac{243}{254} & \frac{127}{117} \cdot 1 & \frac{127}{117} \cdot \frac{114000}{127} \end{array}]$

Step 1.10.2

Simplify $R_{4}$ .

Step 1.11

Perform the row operation $R_{3} = R_{3} - \frac{11}{127} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Step 1.11.1

Perform the row operation $R_{3} = R_{3} - \frac{11}{127} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & \frac{3}{13} & - \frac{2}{13} & 0 & 0 & \frac{1500}{13} \\ 0 - \frac{11}{127} \cdot 0 & 0 - \frac{11}{127} \cdot 0 & 1 - \frac{11}{127} \cdot 0 & \frac{11}{127} - \frac{11}{127} \cdot 1 & \frac{10}{127} - \frac{11}{127} (- \frac{353}{234}) & - \frac{13}{127} - \frac{11}{127} \cdot \frac{27}{26} & 0 - \frac{11}{127} \cdot \frac{127}{117} & \frac{1600}{127} - \frac{11}{127} \cdot \frac{38000}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Step 1.11.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & \frac{3}{13} & - \frac{2}{13} & 0 & 0 & \frac{1500}{13} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Step 1.12

Perform the row operation $R_{2} = R_{2} - \frac{3}{13} R_{4}$ to make the entry at $2, 4$ a $0$ .

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Step 1.12.1

Perform the row operation $R_{2} = R_{2} - \frac{3}{13} R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 - \frac{3}{13} \cdot 0 & 1 - \frac{3}{13} \cdot 0 & \frac{11}{13} - \frac{3}{13} \cdot 0 & \frac{3}{13} - \frac{3}{13} \cdot 1 & - \frac{2}{13} - \frac{3}{13} (- \frac{353}{234}) & 0 - \frac{3}{13} \cdot \frac{27}{26} & 0 - \frac{3}{13} \cdot \frac{127}{117} & \frac{1500}{13} - \frac{3}{13} \cdot \frac{38000}{39} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Step 1.12.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & \frac{1}{4} & 0 & 0 & 0 & 175 \\ 0 & 1 & \frac{11}{13} & 0 & \frac{197}{1014} & - \frac{81}{338} & - \frac{127}{507} & - \frac{18500}{169} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Step 1.13

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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Step 1.13.1

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

$[\begin{array}{c} 1 - \frac{1}{4} \cdot 0 & \frac{5}{4} - \frac{1}{4} \cdot 0 & \frac{9}{4} - \frac{1}{4} \cdot 0 & \frac{1}{4} - \frac{1}{4} \cdot 1 & 0 - \frac{1}{4} (- \frac{353}{234}) & 0 - \frac{1}{4} \cdot \frac{27}{26} & 0 - \frac{1}{4} \cdot \frac{127}{117} & 175 - \frac{1}{4} \cdot \frac{38000}{39} \\ 0 & 1 & \frac{11}{13} & 0 & \frac{197}{1014} & - \frac{81}{338} & - \frac{127}{507} & - \frac{18500}{169} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Step 1.13.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & 0 & \frac{353}{936} & - \frac{27}{104} & - \frac{127}{468} & - \frac{2675}{39} \\ 0 & 1 & \frac{11}{13} & 0 & \frac{197}{1014} & - \frac{81}{338} & - \frac{127}{507} & - \frac{18500}{169} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Step 1.14

Perform the row operation $R_{2} = R_{2} - \frac{11}{13} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 1.14.1

Perform the row operation $R_{2} = R_{2} - \frac{11}{13} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & 0 & \frac{353}{936} & - \frac{27}{104} & - \frac{127}{468} & - \frac{2675}{39} \\ 0 - \frac{11}{13} \cdot 0 & 1 - \frac{11}{13} \cdot 0 & \frac{11}{13} - \frac{11}{13} \cdot 1 & 0 - \frac{11}{13} \cdot 0 & \frac{197}{1014} - \frac{11}{13} \cdot \frac{49}{234} & - \frac{81}{338} - \frac{11}{13} (- \frac{5}{26}) & - \frac{127}{507} - \frac{11}{13} (- \frac{11}{117}) & - \frac{18500}{169} - \frac{11}{13} (- \frac{2800}{39}) \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Step 1.14.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{4} & \frac{9}{4} & 0 & \frac{353}{936} & - \frac{27}{104} & - \frac{127}{468} & - \frac{2675}{39} \\ 0 & 1 & 0 & 0 & \frac{2}{117} & - \frac{1}{13} & - \frac{20}{117} & - \frac{1900}{39} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Step 1.15

Perform the row operation $R_{1} = R_{1} - \frac{9}{4} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Step 1.15.1

Perform the row operation $R_{1} = R_{1} - \frac{9}{4} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{9}{4} \cdot 0 & \frac{5}{4} - \frac{9}{4} \cdot 0 & \frac{9}{4} - \frac{9}{4} \cdot 1 & 0 - \frac{9}{4} \cdot 0 & \frac{353}{936} - \frac{9}{4} \cdot \frac{49}{234} & - \frac{27}{104} - \frac{9}{4} (- \frac{5}{26}) & - \frac{127}{468} - \frac{9}{4} (- \frac{11}{117}) & - \frac{2675}{39} - \frac{9}{4} (- \frac{2800}{39}) \\ 0 & 1 & 0 & 0 & \frac{2}{117} & - \frac{1}{13} & - \frac{20}{117} & - \frac{1900}{39} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Step 1.15.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{5}{4} & 0 & 0 & - \frac{11}{117} & \frac{9}{52} & - \frac{7}{117} & \frac{3625}{39} \\ 0 & 1 & 0 & 0 & \frac{2}{117} & - \frac{1}{13} & - \frac{20}{117} & - \frac{1900}{39} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Step 1.16

Perform the row operation $R_{1} = R_{1} - \frac{5}{4} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 1.16.1

Perform the row operation $R_{1} = R_{1} - \frac{5}{4} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{5}{4} \cdot 0 & \frac{5}{4} - \frac{5}{4} \cdot 1 & 0 - \frac{5}{4} \cdot 0 & 0 - \frac{5}{4} \cdot 0 & - \frac{11}{117} - \frac{5}{4} \cdot \frac{2}{117} & \frac{9}{52} - \frac{5}{4} (- \frac{1}{13}) & - \frac{7}{117} - \frac{5}{4} (- \frac{20}{117}) & \frac{3625}{39} - \frac{5}{4} (- \frac{1900}{39}) \\ 0 & 1 & 0 & 0 & \frac{2}{117} & - \frac{1}{13} & - \frac{20}{117} & - \frac{1900}{39} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Step 1.16.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & - \frac{3}{26} & \frac{7}{26} & \frac{2}{13} & \frac{2000}{13} \\ 0 & 1 & 0 & 0 & \frac{2}{117} & - \frac{1}{13} & - \frac{20}{117} & - \frac{1900}{39} \\ 0 & 0 & 1 & 0 & \frac{49}{234} & - \frac{5}{26} & - \frac{11}{117} & - \frac{2800}{39} \\ 0 & 0 & 0 & 1 & - \frac{353}{234} & \frac{27}{26} & \frac{127}{117} & \frac{38000}{39} \end{array}]$

Step 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 $4$