Linear Algebra Examples

$[\begin{array}{c} 4 & 15 & 4 & 16 & 0 \\ 8 & 1 & 2 & 8 & 0 \\ 5 & 6 & 5 & 20 & 0 \\ 3 & 2 & 2 & 8 & 0 \end{array}]$

Step 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

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

$[\begin{array}{c} \frac{4}{4} & \frac{15}{4} & \frac{4}{4} & \frac{16}{4} & \frac{0}{4} \\ 8 & 1 & 2 & 8 & 0 \\ 5 & 6 & 5 & 20 & 0 \\ 3 & 2 & 2 & 8 & 0 \end{array}]$

Step 1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 8 & 1 & 2 & 8 & 0 \\ 5 & 6 & 5 & 20 & 0 \\ 3 & 2 & 2 & 8 & 0 \end{array}]$

Step 2

Perform the row operation $R_{2} = R_{2} - 8 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 2.1

Perform the row operation $R_{2} = R_{2} - 8 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 8 - 8 \cdot 1 & 1 - 8 (\frac{15}{4}) & 2 - 8 \cdot 1 & 8 - 8 \cdot 4 & 0 - 8 \cdot 0 \\ 5 & 6 & 5 & 20 & 0 \\ 3 & 2 & 2 & 8 & 0 \end{array}]$

Step 2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 0 & - 29 & - 6 & - 24 & 0 \\ 5 & 6 & 5 & 20 & 0 \\ 3 & 2 & 2 & 8 & 0 \end{array}]$

Step 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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Step 3.1

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

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 0 & - 29 & - 6 & - 24 & 0 \\ 5 - 5 \cdot 1 & 6 - 5 (\frac{15}{4}) & 5 - 5 \cdot 1 & 20 - 5 \cdot 4 & 0 - 5 \cdot 0 \\ 3 & 2 & 2 & 8 & 0 \end{array}]$

Step 3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 0 & - 29 & - 6 & - 24 & 0 \\ 0 & - \frac{51}{4} & 0 & 0 & 0 \\ 3 & 2 & 2 & 8 & 0 \end{array}]$

Step 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 4.1

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

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 0 & - 29 & - 6 & - 24 & 0 \\ 0 & - \frac{51}{4} & 0 & 0 & 0 \\ 3 - 3 \cdot 1 & 2 - 3 (\frac{15}{4}) & 2 - 3 \cdot 1 & 8 - 3 \cdot 4 & 0 - 3 \cdot 0 \end{array}]$

Step 4.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 0 & - 29 & - 6 & - 24 & 0 \\ 0 & - \frac{51}{4} & 0 & 0 & 0 \\ 0 & - \frac{37}{4} & - 1 & - 4 & 0 \end{array}]$

Step 5

Multiply each element of $R_{2}$ by $- \frac{1}{29}$ to make the entry at $2, 2$ a $1$ .

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Step 5.1

Multiply each element of $R_{2}$ by $- \frac{1}{29}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ - \frac{1}{29} \cdot 0 & - \frac{1}{29} \cdot - 29 & - \frac{1}{29} \cdot - 6 & - \frac{1}{29} \cdot - 24 & - \frac{1}{29} \cdot 0 \\ 0 & - \frac{51}{4} & 0 & 0 & 0 \\ 0 & - \frac{37}{4} & - 1 & - 4 & 0 \end{array}]$

Step 5.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 0 & 1 & \frac{6}{29} & \frac{24}{29} & 0 \\ 0 & - \frac{51}{4} & 0 & 0 & 0 \\ 0 & - \frac{37}{4} & - 1 & - 4 & 0 \end{array}]$

Step 6

Perform the row operation $R_{3} = R_{3} + \frac{51}{4} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 6.1

Perform the row operation $R_{3} = R_{3} + \frac{51}{4} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 0 & 1 & \frac{6}{29} & \frac{24}{29} & 0 \\ 0 + \frac{51}{4} \cdot 0 & - \frac{51}{4} + \frac{51}{4} \cdot 1 & 0 + \frac{51}{4} \cdot \frac{6}{29} & 0 + \frac{51}{4} \cdot \frac{24}{29} & 0 + \frac{51}{4} \cdot 0 \\ 0 & - \frac{37}{4} & - 1 & - 4 & 0 \end{array}]$

Step 6.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 0 & 1 & \frac{6}{29} & \frac{24}{29} & 0 \\ 0 & 0 & \frac{153}{58} & \frac{306}{29} & 0 \\ 0 & - \frac{37}{4} & - 1 & - 4 & 0 \end{array}]$

Step 7

Perform the row operation $R_{4} = R_{4} + \frac{37}{4} R_{2}$ to make the entry at $4, 2$ a $0$ .

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Step 7.1

Perform the row operation $R_{4} = R_{4} + \frac{37}{4} R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 0 & 1 & \frac{6}{29} & \frac{24}{29} & 0 \\ 0 & 0 & \frac{153}{58} & \frac{306}{29} & 0 \\ 0 + \frac{37}{4} \cdot 0 & - \frac{37}{4} + \frac{37}{4} \cdot 1 & - 1 + \frac{37}{4} \cdot \frac{6}{29} & - 4 + \frac{37}{4} \cdot \frac{24}{29} & 0 + \frac{37}{4} \cdot 0 \end{array}]$

Step 7.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 0 & 1 & \frac{6}{29} & \frac{24}{29} & 0 \\ 0 & 0 & \frac{153}{58} & \frac{306}{29} & 0 \\ 0 & 0 & \frac{53}{58} & \frac{106}{29} & 0 \end{array}]$

Step 8

Multiply each element of $R_{3}$ by $\frac{58}{153}$ to make the entry at $3, 3$ a $1$ .

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Step 8.1

Multiply each element of $R_{3}$ by $\frac{58}{153}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 0 & 1 & \frac{6}{29} & \frac{24}{29} & 0 \\ \frac{58}{153} \cdot 0 & \frac{58}{153} \cdot 0 & \frac{58}{153} \cdot \frac{153}{58} & \frac{58}{153} \cdot \frac{306}{29} & \frac{58}{153} \cdot 0 \\ 0 & 0 & \frac{53}{58} & \frac{106}{29} & 0 \end{array}]$

Step 8.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 0 & 1 & \frac{6}{29} & \frac{24}{29} & 0 \\ 0 & 0 & 1 & 4 & 0 \\ 0 & 0 & \frac{53}{58} & \frac{106}{29} & 0 \end{array}]$

Step 9

Perform the row operation $R_{4} = R_{4} - \frac{53}{58} R_{3}$ to make the entry at $4, 3$ a $0$ .

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Step 9.1

Perform the row operation $R_{4} = R_{4} - \frac{53}{58} R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 0 & 1 & \frac{6}{29} & \frac{24}{29} & 0 \\ 0 & 0 & 1 & 4 & 0 \\ 0 - \frac{53}{58} \cdot 0 & 0 - \frac{53}{58} \cdot 0 & \frac{53}{58} - \frac{53}{58} \cdot 1 & \frac{106}{29} - \frac{53}{58} \cdot 4 & 0 - \frac{53}{58} \cdot 0 \end{array}]$

Step 9.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 0 & 1 & \frac{6}{29} & \frac{24}{29} & 0 \\ 0 & 0 & 1 & 4 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 10

Perform the row operation $R_{2} = R_{2} - \frac{6}{29} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 10.1

Perform the row operation $R_{2} = R_{2} - \frac{6}{29} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 0 - \frac{6}{29} \cdot 0 & 1 - \frac{6}{29} \cdot 0 & \frac{6}{29} - \frac{6}{29} \cdot 1 & \frac{24}{29} - \frac{6}{29} \cdot 4 & 0 - \frac{6}{29} \cdot 0 \\ 0 & 0 & 1 & 4 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 10.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 1 & 4 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 4 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 11

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

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Step 11.1

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

$[\begin{array}{c} 1 - 0 & \frac{15}{4} - 0 & 1 - 1 & 4 - 4 & 0 - 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 4 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 11.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{15}{4} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 4 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 12

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

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Step 12.1

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

$[\begin{array}{c} 1 - \frac{15}{4} \cdot 0 & \frac{15}{4} - \frac{15}{4} \cdot 1 & 0 - \frac{15}{4} \cdot 0 & 0 - \frac{15}{4} \cdot 0 & 0 - \frac{15}{4} \cdot 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 4 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 12.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 4 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$