Algèbre linéaire Exemples

{(2 A^{T})}^{- 1} = [\begin{array}{c} 4 & 5 \\ 3 & 4 \end{array}]

${(2 A^{T})}^{- 1} = [\begin{array}{c} 4 & 5 \\ 3 & 4 \end{array}]$

Étape 1

Déterminez la forme d’échelon en ligne réduite.

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Étape 1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{4}

$\frac{1}{4}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

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Étape 1.1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{4}

$\frac{1}{4}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

[\begin{array}{c} \frac{4}{4} & \frac{5}{4} \\ 3 & 4 \end{array}]

$[\begin{array}{c} \frac{4}{4} & \frac{5}{4} \\ 3 & 4 \end{array}]$

Étape 1.1.2

Simplifiez

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & \frac{5}{4} \\ 3 & 4 \end{array}]

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

[\begin{array}{c} 1 & \frac{5}{4} \\ 3 & 4 \end{array}]

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

Étape 1.2

Perform the row operation

R_{2} = R_{2} - 3 R_{1}

$R_{2} = R_{2} - 3 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Étape 1.2.1

Perform the row operation

R_{2} = R_{2} - 3 R_{1}

$R_{2} = R_{2} - 3 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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

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

Étape 1.2.2

Simplifiez

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & \frac{5}{4} \\ 0 & \frac{1}{4} \end{array}]

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

[\begin{array}{c} 1 & \frac{5}{4} \\ 0 & \frac{1}{4} \end{array}]

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

Étape 1.3

Multiply each element of

R_{2}

$R_{2}$ by

4

$4$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Étape 1.3.1

Multiply each element of

R_{2}

$R_{2}$ by

4

$4$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

[\begin{array}{c} 1 & \frac{5}{4} \\ 4 \cdot 0 & 4 (\frac{1}{4}) \end{array}]

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

Étape 1.3.2

Simplifiez

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & \frac{5}{4} \\ 0 & 1 \end{array}]

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

[\begin{array}{c} 1 & \frac{5}{4} \\ 0 & 1 \end{array}]

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

Étape 1.4

Perform the row operation

R_{1} = R_{1} - \frac{5}{4} R_{2}

$R_{1} = R_{1} - \frac{5}{4} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

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Étape 1.4.1

Perform the row operation

R_{1} = R_{1} - \frac{5}{4} R_{2}

$R_{1} = R_{1} - \frac{5}{4} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

[\begin{array}{c} 1 - \frac{5}{4} \cdot 0 & \frac{5}{4} - \frac{5}{4} \cdot 1 \\ 0 & 1 \end{array}]

$[\begin{array}{c} 1 - \frac{5}{4} \cdot 0 & \frac{5}{4} - \frac{5}{4} \cdot 1 \\ 0 & 1 \end{array}]$

Étape 1.4.2

Simplifiez

R_{1}

$R_{1}$ .

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

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

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

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

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

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

Étape 2

The pivot positions are the locations with the leading

1

$1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions:

a_{11}

$a_{11}$ and

a_{22}

$a_{22}$

Pivot Columns:

1

$1$ and

2

$2$

Étape 3

The rank is the number of pivot columns.

2

$2$