Linear Algebra Examples

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0.5 & 0.33 & 0.25 & 0.21 & 1.2 \\ 0.33 & 0.25 & 0.2 & 0.17 & 1.3 \\ 0.25 & 0.2 & 0.17 & 0.14 & 1.4 \end{array}]$

Step 1

Perform the row operation $R_{2} = R_{2} - 0.5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 1.1

Perform the row operation $R_{2} = R_{2} - 0.5 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0.5 - 0.5 \cdot 1 & 0.33 - 0.5 \cdot 0.5 & 0.25 - 0.5 \cdot 0.33 & 0.21 - 0.5 \cdot 0.25 & 1.2 - 0.5 \cdot 1.1 \\ 0.33 & 0.25 & 0.2 & 0.17 & 1.3 \\ 0.25 & 0.2 & 0.17 & 0.14 & 1.4 \end{array}]$

Step 1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 0.08 & 0.085 & 0.085 & 0.65 \\ 0.33 & 0.25 & 0.2 & 0.17 & 1.3 \\ 0.25 & 0.2 & 0.17 & 0.14 & 1.4 \end{array}]$

Step 2

Perform the row operation $R_{3} = R_{3} - 0.33 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 2.1

Perform the row operation $R_{3} = R_{3} - 0.33 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 0.08 & 0.085 & 0.085 & 0.65 \\ 0.33 - 0.33 \cdot 1 & 0.25 - 0.33 \cdot 0.5 & 0.2 - 0.33 \cdot 0.33 & 0.17 - 0.33 \cdot 0.25 & 1.3 - 0.33 \cdot 1.1 \\ 0.25 & 0.2 & 0.17 & 0.14 & 1.4 \end{array}]$

Step 2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 0.08 & 0.085 & 0.085 & 0.65 \\ 0 & 0.085 & 0.0911 & 0.0875 & 0.937 \\ 0.25 & 0.2 & 0.17 & 0.14 & 1.4 \end{array}]$

Step 3

Perform the row operation $R_{4} = R_{4} - 0.25 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Step 3.1

Perform the row operation $R_{4} = R_{4} - 0.25 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 0.08 & 0.085 & 0.085 & 0.65 \\ 0 & 0.085 & 0.0911 & 0.0875 & 0.937 \\ 0.25 - 0.25 \cdot 1 & 0.2 - 0.25 \cdot 0.5 & 0.17 - 0.25 \cdot 0.33 & 0.14 - 0.25 \cdot 0.25 & 1.4 - 0.25 \cdot 1.1 \end{array}]$

Step 3.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 0.08 & 0.085 & 0.085 & 0.65 \\ 0 & 0.085 & 0.0911 & 0.0875 & 0.937 \\ 0 & 0.075 & 0.0875 & 0.0775 & 1.125 \end{array}]$

Step 4

Multiply each element of $R_{2}$ by $\frac{1}{0.08}$ to make the entry at $2, 2$ a $1$ .

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Step 4.1

Multiply each element of $R_{2}$ by $\frac{1}{0.08}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ \frac{0}{0.08} & \frac{0.08}{0.08} & \frac{0.085}{0.08} & \frac{0.085}{0.08} & \frac{0.65}{0.08} \\ 0 & 0.085 & 0.0911 & 0.0875 & 0.937 \\ 0 & 0.075 & 0.0875 & 0.0775 & 1.125 \end{array}]$

Step 4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 1 & 1.0625 & 1.0625 & 8.125 \\ 0 & 0.085 & 0.0911 & 0.0875 & 0.937 \\ 0 & 0.075 & 0.0875 & 0.0775 & 1.125 \end{array}]$

Step 5

Perform the row operation $R_{3} = R_{3} - 0.085 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 5.1

Perform the row operation $R_{3} = R_{3} - 0.085 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 1 & 1.0625 & 1.0625 & 8.125 \\ 0 - 0.085 \cdot 0 & 0.085 - 0.085 \cdot 1 & 0.0911 - 0.085 \cdot 1.0625 & 0.0875 - 0.085 \cdot 1.0625 & 0.937 - 0.085 \cdot 8.125 \\ 0 & 0.075 & 0.0875 & 0.0775 & 1.125 \end{array}]$

Step 5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 1 & 1.0625 & 1.0625 & 8.125 \\ 0 & 0 & 0.0007875 & - 0.0028125 & 0.246375 \\ 0 & 0.075 & 0.0875 & 0.0775 & 1.125 \end{array}]$

Step 6

Perform the row operation $R_{4} = R_{4} - 0.075 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Step 6.1

Perform the row operation $R_{4} = R_{4} - 0.075 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 1 & 1.0625 & 1.0625 & 8.125 \\ 0 & 0 & 0.0007875 & - 0.0028125 & 0.246375 \\ 0 - 0.075 \cdot 0 & 0.075 - 0.075 \cdot 1 & 0.0875 - 0.075 \cdot 1.0625 & 0.0775 - 0.075 \cdot 1.0625 & 1.125 - 0.075 \cdot 8.125 \end{array}]$

Step 6.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 1 & 1.0625 & 1.0625 & 8.125 \\ 0 & 0 & 0.0007875 & - 0.0028125 & 0.246375 \\ 0 & 0 & 0.0078125 & - 0.0021875 & 0.515625 \end{array}]$

Step 7

Multiply each element of $R_{3}$ by $\frac{1}{0.0007875}$ to make the entry at $3, 3$ a $1$ .

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Step 7.1

Multiply each element of $R_{3}$ by $\frac{1}{0.0007875}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 1 & 1.0625 & 1.0625 & 8.125 \\ \frac{0}{0.0007875} & \frac{0}{0.0007875} & \frac{0.0007875}{0.0007875} & \frac{- 0.0028125}{0.0007875} & \frac{0.246375}{0.0007875} \\ 0 & 0 & 0.0078125 & - 0.0021875 & 0.515625 \end{array}]$

Step 7.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 1 & 1.0625 & 1.0625 & 8.125 \\ 0 & 0 & 1 & - 3.\overline{571428} & 312.\overline{857142} \\ 0 & 0 & 0.0078125 & - 0.0021875 & 0.515625 \end{array}]$

Step 8

Perform the row operation $R_{4} = R_{4} - 0.0078125 R_{3}$ to make the entry at $4, 3$ a $0$ .

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Step 8.1

Perform the row operation $R_{4} = R_{4} - 0.0078125 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 1 & 1.0625 & 1.0625 & 8.125 \\ 0 & 0 & 1 & - 3.\overline{571428} & 312.\overline{857142} \\ 0 - 0.0078125 \cdot 0 & 0 - 0.0078125 \cdot 0 & 0.0078125 - 0.0078125 \cdot 1 & - 0.0021875 - 0.0078125 \cdot - 3.\overline{571428} & 0.515625 - 0.0078125 \cdot 312.\overline{857142} \end{array}]$

Step 8.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 1 & 1.0625 & 1.0625 & 8.125 \\ 0 & 0 & 1 & - 3.\overline{571428} & 312.\overline{857142} \\ 0 & 0 & 0 & 0.02571428 & - 1.9\overline{285714} \end{array}]$

Step 9

Multiply each element of $R_{4}$ by $\frac{1}{0.02571428}$ to make the entry at $4, 4$ a $1$ .

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Step 9.1

Multiply each element of $R_{4}$ by $\frac{1}{0.02571428}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 1 & 1.0625 & 1.0625 & 8.125 \\ 0 & 0 & 1 & - 3.\overline{571428} & 312.\overline{857142} \\ \frac{0}{0.02571428} & \frac{0}{0.02571428} & \frac{0}{0.02571428} & \frac{0.02571428}{0.02571428} & \frac{- 1.9\overline{285714}}{0.02571428} \end{array}]$

Step 9.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 1 & 1.0625 & 1.0625 & 8.125 \\ 0 & 0 & 1 & - 3.\overline{571428} & 312.\overline{857142} \\ 0 & 0 & 0 & 1 & - 75 \end{array}]$

Step 10

Perform the row operation $R_{3} = R_{3} + 3.\overline{571428} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Step 10.1

Perform the row operation $R_{3} = R_{3} + 3.\overline{571428} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 1 & 1.0625 & 1.0625 & 8.125 \\ 0 + 3.\overline{571428} \cdot 0 & 0 + 3.\overline{571428} \cdot 0 & 1 + 3.\overline{571428} \cdot 0 & - 3.\overline{571428} + 3.\overline{571428} \cdot 1 & 312.\overline{857142} + 3.\overline{571428} \cdot - 75 \\ 0 & 0 & 0 & 1 & - 75 \end{array}]$

Step 10.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 1 & 1.0625 & 1.0625 & 8.125 \\ 0 & 0 & 1 & 0 & 45 \\ 0 & 0 & 0 & 1 & - 75 \end{array}]$

Step 11

Perform the row operation $R_{2} = R_{2} - 1.0625 R_{4}$ to make the entry at $2, 4$ a $0$ .

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Step 11.1

Perform the row operation $R_{2} = R_{2} - 1.0625 R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 - 1.0625 \cdot 0 & 1 - 1.0625 \cdot 0 & 1.0625 - 1.0625 \cdot 0 & 1.0625 - 1.0625 \cdot 1 & 8.125 - 1.0625 \cdot - 75 \\ 0 & 0 & 1 & 0 & 45 \\ 0 & 0 & 0 & 1 & - 75 \end{array}]$

Step 11.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0.25 & 1.1 \\ 0 & 1 & 1.0625 & 0 & 87.8125 \\ 0 & 0 & 1 & 0 & 45 \\ 0 & 0 & 0 & 1 & - 75 \end{array}]$

Step 12

Perform the row operation $R_{1} = R_{1} - 0.25 R_{4}$ to make the entry at $1, 4$ a $0$ .

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Step 12.1

Perform the row operation $R_{1} = R_{1} - 0.25 R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - 0.25 \cdot 0 & 0.5 - 0.25 \cdot 0 & 0.33 - 0.25 \cdot 0 & 0.25 - 0.25 \cdot 1 & 1.1 - 0.25 \cdot - 75 \\ 0 & 1 & 1.0625 & 0 & 87.8125 \\ 0 & 0 & 1 & 0 & 45 \\ 0 & 0 & 0 & 1 & - 75 \end{array}]$

Step 12.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0 & 19.85 \\ 0 & 1 & 1.0625 & 0 & 87.8125 \\ 0 & 0 & 1 & 0 & 45 \\ 0 & 0 & 0 & 1 & - 75 \end{array}]$

Step 13

Perform the row operation $R_{2} = R_{2} - 1.0625 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 13.1

Perform the row operation $R_{2} = R_{2} - 1.0625 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0 & 19.85 \\ 0 - 1.0625 \cdot 0 & 1 - 1.0625 \cdot 0 & 1.0625 - 1.0625 \cdot 1 & 0 - 1.0625 \cdot 0 & 87.8125 - 1.0625 \cdot 45 \\ 0 & 0 & 1 & 0 & 45 \\ 0 & 0 & 0 & 1 & - 75 \end{array}]$

Step 13.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0.5 & 0.33 & 0 & 19.85 \\ 0 & 1 & 0 & 0 & 40 \\ 0 & 0 & 1 & 0 & 45 \\ 0 & 0 & 0 & 1 & - 75 \end{array}]$

Step 14

Perform the row operation $R_{1} = R_{1} - 0.33 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Step 14.1

Perform the row operation $R_{1} = R_{1} - 0.33 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 0.33 \cdot 0 & 0.5 - 0.33 \cdot 0 & 0.33 - 0.33 \cdot 1 & 0 - 0.33 \cdot 0 & 19.85 - 0.33 \cdot 45 \\ 0 & 1 & 0 & 0 & 40 \\ 0 & 0 & 1 & 0 & 45 \\ 0 & 0 & 0 & 1 & - 75 \end{array}]$

Step 14.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0.5 & 0 & 0 & 5 \\ 0 & 1 & 0 & 0 & 40 \\ 0 & 0 & 1 & 0 & 45 \\ 0 & 0 & 0 & 1 & - 75 \end{array}]$

Step 15

Perform the row operation $R_{1} = R_{1} - 0.5 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 15.1

Perform the row operation $R_{1} = R_{1} - 0.5 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 0.5 \cdot 0 & 0.5 - 0.5 \cdot 1 & 0 - 0.5 \cdot 0 & 0 - 0.5 \cdot 0 & 5 - 0.5 \cdot 40 \\ 0 & 1 & 0 & 0 & 40 \\ 0 & 0 & 1 & 0 & 45 \\ 0 & 0 & 0 & 1 & - 75 \end{array}]$

Step 15.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & - 15 \\ 0 & 1 & 0 & 0 & 40 \\ 0 & 0 & 1 & 0 & 45 \\ 0 & 0 & 0 & 1 & - 75 \end{array}]$