Algebra lineare Esempi

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 2 & 5 3 & 0 \end{matrix} ⎤ ⎥ ⎦

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

Passaggio 1

Perform the row operation

R_{2} = R_{2} - 2 R_{1}

$R_{2} = R_{2} - 2 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Passaggio 1.1

Perform the row operation

R_{2} = R_{2} - 2 R_{1}

$R_{2} = R_{2} - 2 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 2 - 2 \cdot 1 & 5 - 2 \cdot 2 3 & 0 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 2 \\ 2 - 2 \cdot 1 & 5 - 2 \cdot 2 \\ 3 & 0 \end{array}]$

Passaggio 1.2

Semplifica

R_{2}

$R_{2}$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 0 & 1 3 & 0 \end{matrix} ⎤ ⎥ ⎦

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

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 0 & 1 3 & 0 \end{matrix} ⎤ ⎥ ⎦

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

Passaggio 2

Perform the row operation

R_{3} = R_{3} - 3 R_{1}

$R_{3} = R_{3} - 3 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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Passaggio 2.1

Perform the row operation

R_{3} = R_{3} - 3 R_{1}

$R_{3} = R_{3} - 3 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 0 & 1 3 - 3 \cdot 1 & 0 - 3 \cdot 2 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 2 \\ 0 & 1 \\ 3 - 3 \cdot 1 & 0 - 3 \cdot 2 \end{array}]$

Passaggio 2.2

Semplifica

R_{3}

$R_{3}$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 0 & 1 0 & - 6 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 2 \\ 0 & 1 \\ 0 & - 6 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 0 & 1 0 & - 6 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 2 \\ 0 & 1 \\ 0 & - 6 \end{array}]$

Passaggio 3

Perform the row operation

R_{3} = R_{3} + 6 R_{2}

$R_{3} = R_{3} + 6 R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

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Passaggio 3.1

Perform the row operation

R_{3} = R_{3} + 6 R_{2}

$R_{3} = R_{3} + 6 R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 0 & 1 0 + 6 \cdot 0 & - 6 + 6 \cdot 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 2 \\ 0 & 1 \\ 0 + 6 \cdot 0 & - 6 + 6 \cdot 1 \end{array}]$

Passaggio 3.2

Semplifica

R_{3}

$R_{3}$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 0 & 1 0 & 0 \end{matrix} ⎤ ⎥ ⎦

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

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 0 & 1 0 & 0 \end{matrix} ⎤ ⎥ ⎦

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

Passaggio 4

Perform the row operation

R_{1} = R_{1} - 2 R_{2}

$R_{1} = R_{1} - 2 R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

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Passaggio 4.1

Perform the row operation

R_{1} = R_{1} - 2 R_{2}

$R_{1} = R_{1} - 2 R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

⎡ ⎢ ⎣ \begin{matrix} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 0 & 1 0 & 0 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 \\ 0 & 1 \\ 0 & 0 \end{array}]$

Passaggio 4.2

Semplifica

R_{1}

$R_{1}$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 0 & 1 0 & 0 \end{matrix} ⎤ ⎥ ⎦

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

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 0 & 1 0 & 0 \end{matrix} ⎤ ⎥ ⎦

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

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