Algèbre linéaire Exemples

[\begin{array}{c} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}]$

Étape 1

Perform the row operation

R_{2} = R_{2} - 4 R_{1}

$R_{2} = R_{2} - 4 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Étape 1.1

Perform the row operation

R_{2} = R_{2} - 4 R_{1}

$R_{2} = R_{2} - 4 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

[\begin{array}{c} 1 & 2 & 3 \\ 4 - 4 \cdot 1 & 5 - 4 \cdot 2 & 6 - 4 \cdot 3 \\ 7 & 8 & 9 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 4 - 4 \cdot 1 & 5 - 4 \cdot 2 & 6 - 4 \cdot 3 \\ 7 & 8 & 9 \end{array}]$

Étape 1.2

Simplifiez

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 3 & - 6 \\ 7 & 8 & 9 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 3 & - 6 \\ 7 & 8 & 9 \end{array}]$

[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 3 & - 6 \\ 7 & 8 & 9 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 3 & - 6 \\ 7 & 8 & 9 \end{array}]$

Étape 2

Perform the row operation

R_{3} = R_{3} - 7 R_{1}

$R_{3} = R_{3} - 7 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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Étape 2.1

Perform the row operation

R_{3} = R_{3} - 7 R_{1}

$R_{3} = R_{3} - 7 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 3 & - 6 \\ 7 - 7 \cdot 1 & 8 - 7 \cdot 2 & 9 - 7 \cdot 3 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 3 & - 6 \\ 7 - 7 \cdot 1 & 8 - 7 \cdot 2 & 9 - 7 \cdot 3 \end{array}]$

Étape 2.2

Simplifiez

R_{3}

$R_{3}$ .

[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 3 & - 6 \\ 0 & - 6 & - 12 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 3 & - 6 \\ 0 & - 6 & - 12 \end{array}]$

[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 3 & - 6 \\ 0 & - 6 & - 12 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 3 & - 6 \\ 0 & - 6 & - 12 \end{array}]$

Étape 3

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{1}{3}

$- \frac{1}{3}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Étape 3.1

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{1}{3}

$- \frac{1}{3}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

[\begin{array}{c} 1 & 2 & 3 \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot - 6 \\ 0 & - 6 & - 12 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot - 6 \\ 0 & - 6 & - 12 \end{array}]$

Étape 3.2

Simplifiez

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & - 6 & - 12 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & - 6 & - 12 \end{array}]$

[\begin{array}{c} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & - 6 & - 12 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & - 6 & - 12 \end{array}]$

Étape 4

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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Étape 4.1

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{array}{c} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 + 6 \cdot 0 & - 6 + 6 \cdot 1 & - 12 + 6 \cdot 2 \end{array}]

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 + 6 \cdot 0 & - 6 + 6 \cdot 1 & - 12 + 6 \cdot 2 \end{array}]$

Étape 4.2

Simplifiez

R_{3}

$R_{3}$ .

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

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

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

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

Étape 5

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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Étape 5.1

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

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

Étape 5.2

Simplifiez

R_{1}

$R_{1}$ .

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

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

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

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