Algebra lineare Esempi

$[\begin{array}{c} 4 & 20 & 31 \\ 6 & - 5 & - 6 \\ 2 & - 11 & - 16 \end{array}]$

Passaggio 1

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

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Passaggio 1.1

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

$[\begin{array}{c} \frac{4}{4} & \frac{20}{4} & \frac{31}{4} \\ 6 & - 5 & - 6 \\ 2 & - 11 & - 16 \end{array}]$

Passaggio 1.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 5 & \frac{31}{4} \\ 6 & - 5 & - 6 \\ 2 & - 11 & - 16 \end{array}]$

Passaggio 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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Passaggio 2.1

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

$[\begin{array}{c} 1 & 5 & \frac{31}{4} \\ 6 - 6 \cdot 1 & - 5 - 6 \cdot 5 & - 6 - 6 (\frac{31}{4}) \\ 2 & - 11 & - 16 \end{array}]$

Passaggio 2.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & 5 & \frac{31}{4} \\ 0 & - 35 & - \frac{105}{2} \\ 2 & - 11 & - 16 \end{array}]$

Passaggio 3

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

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Passaggio 3.1

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

$[\begin{array}{c} 1 & 5 & \frac{31}{4} \\ 0 & - 35 & - \frac{105}{2} \\ 2 - 2 \cdot 1 & - 11 - 2 \cdot 5 & - 16 - 2 (\frac{31}{4}) \end{array}]$

Passaggio 3.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 5 & \frac{31}{4} \\ 0 & - 35 & - \frac{105}{2} \\ 0 & - 21 & - \frac{63}{2} \end{array}]$

Passaggio 4

Multiply each element of $R_{2}$ by $- \frac{1}{35}$ to make the entry at $2, 2$ a $1$ .

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Passaggio 4.1

Multiply each element of $R_{2}$ by $- \frac{1}{35}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 5 & \frac{31}{4} \\ - \frac{1}{35} \cdot 0 & - \frac{1}{35} \cdot - 35 & - \frac{1}{35} (- \frac{105}{2}) \\ 0 & - 21 & - \frac{63}{2} \end{array}]$

Passaggio 4.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & 5 & \frac{31}{4} \\ 0 & 1 & \frac{3}{2} \\ 0 & - 21 & - \frac{63}{2} \end{array}]$

Passaggio 5

Perform the row operation $R_{3} = R_{3} + 21 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Passaggio 5.1

Perform the row operation $R_{3} = R_{3} + 21 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 5 & \frac{31}{4} \\ 0 & 1 & \frac{3}{2} \\ 0 + 21 \cdot 0 & - 21 + 21 \cdot 1 & - \frac{63}{2} + 21 (\frac{3}{2}) \end{array}]$

Passaggio 5.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 5 & \frac{31}{4} \\ 0 & 1 & \frac{3}{2} \\ 0 & 0 & 0 \end{array}]$

Passaggio 6

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

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Passaggio 6.1

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

$[\begin{array}{c} 1 - 5 \cdot 0 & 5 - 5 \cdot 1 & \frac{31}{4} - 5 (\frac{3}{2}) \\ 0 & 1 & \frac{3}{2} \\ 0 & 0 & 0 \end{array}]$

Passaggio 6.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{4} \\ 0 & 1 & \frac{3}{2} \\ 0 & 0 & 0 \end{array}]$