Precalcolo Esempi

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

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

Passaggio 1

Find the determinant.

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Passaggio 1.1

Choose the row or column with the most

0

$0$ elements. If there are no

0

$0$ elements choose any row or column. Multiply every element in column

1

$1$ by its cofactor and add.

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Passaggio 1.1.1

Consider the corresponding sign chart.

| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Passaggio 1.1.2

The cofactor is the minor with the sign changed if the indices match a

-

$-$ position on the sign chart.

Passaggio 1.1.3

The minor for

a_{11}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

| \begin{array}{c} 2 & 0 \\ 3 & 2 \end{array} |

$| \begin{array}{c} 2 & 0 \\ 3 & 2 \end{array} |$

Passaggio 1.1.4

Multiply element

a_{11}

$a_{11}$ by its cofactor.

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

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

Passaggio 1.1.5

The minor for

a_{21}

$a_{21}$ is the determinant with row

2

$2$ and column

1

$1$ deleted.

| \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} |

$| \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} |$

Passaggio 1.1.6

Multiply element

a_{21}

$a_{21}$ by its cofactor.

- 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} |

$- 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} |$

Passaggio 1.1.7

The minor for

a_{31}

$a_{31}$ is the determinant with row

3

$3$ and column

1

$1$ deleted.

| \begin{array}{c} 2 & 2 \\ 2 & 0 \end{array} |

$| \begin{array}{c} 2 & 2 \\ 2 & 0 \end{array} |$

Passaggio 1.1.8

Multiply element

a_{31}

$a_{31}$ by its cofactor.

0 | \begin{array}{c} 2 & 2 \\ 2 & 0 \end{array} |

$0 | \begin{array}{c} 2 & 2 \\ 2 & 0 \end{array} |$

Passaggio 1.1.9

Add the terms together.

1 | \begin{array}{c} 2 & 0 \\ 3 & 2 \end{array} | - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0 | \begin{array}{c} 2 & 2 \\ 2 & 0 \end{array} |

$1 | \begin{array}{c} 2 & 0 \\ 3 & 2 \end{array} | - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0 | \begin{array}{c} 2 & 2 \\ 2 & 0 \end{array} |$

1 | \begin{array}{c} 2 & 0 \\ 3 & 2 \end{array} | - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0 | \begin{array}{c} 2 & 2 \\ 2 & 0 \end{array} |

$1 | \begin{array}{c} 2 & 0 \\ 3 & 2 \end{array} | - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0 | \begin{array}{c} 2 & 2 \\ 2 & 0 \end{array} |$

Passaggio 1.2

Moltiplica

0

$0$ per

| \begin{array}{c} 2 & 2 \\ 2 & 0 \end{array} |

$| \begin{array}{c} 2 & 2 \\ 2 & 0 \end{array} |$ .

1 | \begin{array}{c} 2 & 0 \\ 3 & 2 \end{array} | - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0

$1 | \begin{array}{c} 2 & 0 \\ 3 & 2 \end{array} | - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0$

Passaggio 1.3

Calcola

| \begin{array}{c} 2 & 0 \\ 3 & 2 \end{array} |

$| \begin{array}{c} 2 & 0 \\ 3 & 2 \end{array} |$ .

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Passaggio 1.3.1

È possibile trovare il determinante di una matrice

2 \times 2

$2 \times 2$ usando la formula

| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

1 (2 \cdot 2 - 3 \cdot 0) - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0

$1 (2 \cdot 2 - 3 \cdot 0) - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0$

Passaggio 1.3.2

Semplifica il determinante.

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Passaggio 1.3.2.1

Semplifica ciascun termine.

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Passaggio 1.3.2.1.1

Moltiplica

2

$2$ per

2

$2$ .

1 (4 - 3 \cdot 0) - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0

$1 (4 - 3 \cdot 0) - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0$

Passaggio 1.3.2.1.2

Moltiplica

- 3

$- 3$ per

0

$0$ .

1 (4 + 0) - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0

$1 (4 + 0) - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0$

1 (4 + 0) - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0

$1 (4 + 0) - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0$

Passaggio 1.3.2.2

Somma

4

$4$ e

0

$0$ .

1 \cdot 4 - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0

$1 \cdot 4 - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0$

1 \cdot 4 - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0

$1 \cdot 4 - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0$

1 \cdot 4 - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0

$1 \cdot 4 - 2 | \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} | + 0$

Passaggio 1.4

Calcola

| \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} |

$| \begin{array}{c} 2 & 2 \\ 3 & 2 \end{array} |$ .

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Passaggio 1.4.1

È possibile trovare il determinante di una matrice

2 \times 2

$2 \times 2$ usando la formula

| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

1 \cdot 4 - 2 (2 \cdot 2 - 3 \cdot 2) + 0

$1 \cdot 4 - 2 (2 \cdot 2 - 3 \cdot 2) + 0$

Passaggio 1.4.2

Semplifica il determinante.

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Passaggio 1.4.2.1

Semplifica ciascun termine.

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Passaggio 1.4.2.1.1

Moltiplica

2

$2$ per

2

$2$ .

1 \cdot 4 - 2 (4 - 3 \cdot 2) + 0

$1 \cdot 4 - 2 (4 - 3 \cdot 2) + 0$

Passaggio 1.4.2.1.2

Moltiplica

- 3

$- 3$ per

2

$2$ .

1 \cdot 4 - 2 (4 - 6) + 0

$1 \cdot 4 - 2 (4 - 6) + 0$

1 \cdot 4 - 2 (4 - 6) + 0

$1 \cdot 4 - 2 (4 - 6) + 0$

Passaggio 1.4.2.2

Sottrai

6

$6$ da

4

$4$ .

1 \cdot 4 - 2 \cdot - 2 + 0

$1 \cdot 4 - 2 \cdot - 2 + 0$

1 \cdot 4 - 2 \cdot - 2 + 0

$1 \cdot 4 - 2 \cdot - 2 + 0$

1 \cdot 4 - 2 \cdot - 2 + 0

$1 \cdot 4 - 2 \cdot - 2 + 0$

Passaggio 1.5

Semplifica il determinante.

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Passaggio 1.5.1

Semplifica ciascun termine.

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Passaggio 1.5.1.1

Moltiplica

4

$4$ per

1

$1$ .

4 - 2 \cdot - 2 + 0

$4 - 2 \cdot - 2 + 0$

Passaggio 1.5.1.2

Moltiplica

- 2

$- 2$ per

- 2

$- 2$ .

4 + 4 + 0

$4 + 4 + 0$

4 + 4 + 0

$4 + 4 + 0$

Passaggio 1.5.2

Somma

4

$4$ e

4

$4$ .

8 + 0

$8 + 0$

Passaggio 1.5.3

Somma

8

$8$ e

0

$0$ .

8

$8$

8

$8$

8

$8$

Passaggio 2

Since the determinant is non-zero, the inverse exists.

Passaggio 3

Set up a

3 \times 6

$3 \times 6$ matrix where the left half is the original matrix and the right half is its identity matrix.

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

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

Passaggio 4

Trova la forma ridotta a scala per righe di Echelon.

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Passaggio 4.1

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

$0$ .

Tocca per altri passaggi...

Passaggio 4.1.1

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

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

Passaggio 4.1.2

Semplifica

R_{2}

$R_{2}$ .

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

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

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

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

Passaggio 4.2

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{1}{2}

$- \frac{1}{2}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Passaggio 4.2.1

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{1}{2}

$- \frac{1}{2}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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

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

Passaggio 4.2.2

Semplifica

R_{2}

$R_{2}$ .

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

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

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

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

Passaggio 4.3

Perform the row operation

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

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

3, 2

$3, 2$ a

0

$0$ .

Tocca per altri passaggi...

Passaggio 4.3.1

Perform the row operation

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

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

3, 2

$3, 2$ a

0

$0$ .

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

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

Passaggio 4.3.2

Semplifica

R_{3}

$R_{3}$ .

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

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

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

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

Passaggio 4.4

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{1}{4}

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

3, 3

$3, 3$ a

1

$1$ .

Tocca per altri passaggi...

Passaggio 4.4.1

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{1}{4}

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

3, 3

$3, 3$ a

1

$1$ .

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

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

Passaggio 4.4.2

Semplifica

R_{3}

$R_{3}$ .

[\begin{array}{c} 1 & 2 & 2 & 1 & 0 & 0 \\ 0 & 1 & 2 & 1 & - \frac{1}{2} & 0 \\ 0 & 0 & 1 & \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]

$[\begin{array}{c} 1 & 2 & 2 & 1 & 0 & 0 \\ 0 & 1 & 2 & 1 & - \frac{1}{2} & 0 \\ 0 & 0 & 1 & \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]$

[\begin{array}{c} 1 & 2 & 2 & 1 & 0 & 0 \\ 0 & 1 & 2 & 1 & - \frac{1}{2} & 0 \\ 0 & 0 & 1 & \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]

$[\begin{array}{c} 1 & 2 & 2 & 1 & 0 & 0 \\ 0 & 1 & 2 & 1 & - \frac{1}{2} & 0 \\ 0 & 0 & 1 & \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]$

Passaggio 4.5

Perform the row operation

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

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

2, 3

$2, 3$ a

0

$0$ .

Tocca per altri passaggi...

Passaggio 4.5.1

Perform the row operation

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

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

2, 3

$2, 3$ a

0

$0$ .

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

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

Passaggio 4.5.2

Semplifica

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & 2 & 2 & 1 & 0 & 0 \\ 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{4} & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]

$[\begin{array}{c} 1 & 2 & 2 & 1 & 0 & 0 \\ 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{4} & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]$

[\begin{array}{c} 1 & 2 & 2 & 1 & 0 & 0 \\ 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{4} & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]

$[\begin{array}{c} 1 & 2 & 2 & 1 & 0 & 0 \\ 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{4} & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]$

Passaggio 4.6

Perform the row operation

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

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

1, 3

$1, 3$ a

0

$0$ .

Tocca per altri passaggi...

Passaggio 4.6.1

Perform the row operation

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

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

1, 3

$1, 3$ a

0

$0$ .

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

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

Passaggio 4.6.2

Semplifica

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & 2 & 0 & - \frac{1}{2} & \frac{3}{4} & \frac{1}{2} \\ 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{4} & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]

$[\begin{array}{c} 1 & 2 & 0 & - \frac{1}{2} & \frac{3}{4} & \frac{1}{2} \\ 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{4} & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]$

[\begin{array}{c} 1 & 2 & 0 & - \frac{1}{2} & \frac{3}{4} & \frac{1}{2} \\ 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{4} & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]

Passaggio 4.7

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

$0$ .

Tocca per altri passaggi...

Passaggio 4.7.1

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 & 0 - 2 \cdot 0 & - \frac{1}{2} - 2 (- \frac{1}{2}) & \frac{3}{4} - 2 (\frac{1}{4}) & \frac{1}{2} - 2 (\frac{1}{2}) \\ 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{4} & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]

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

Passaggio 4.7.2

Semplifica

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & 0 & 0 & \frac{1}{2} & \frac{1}{4} & - \frac{1}{2} \\ 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{4} & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]

$[\begin{array}{c} 1 & 0 & 0 & \frac{1}{2} & \frac{1}{4} & - \frac{1}{2} \\ 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{4} & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]$

[\begin{array}{c} 1 & 0 & 0 & \frac{1}{2} & \frac{1}{4} & - \frac{1}{2} \\ 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{4} & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]

[\begin{array}{c} 1 & 0 & 0 & \frac{1}{2} & \frac{1}{4} & - \frac{1}{2} \\ 0 & 1 & 0 & - \frac{1}{2} & \frac{1}{4} & \frac{1}{2} \\ 0 & 0 & 1 & \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]

Passaggio 5

The right half of the reduced row echelon form is the inverse.

[\begin{array}{c} \frac{1}{2} & \frac{1}{4} & - \frac{1}{2} \\ - \frac{1}{2} & \frac{1}{4} & \frac{1}{2} \\ \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]

$[\begin{array}{c} \frac{1}{2} & \frac{1}{4} & - \frac{1}{2} \\ - \frac{1}{2} & \frac{1}{4} & \frac{1}{2} \\ \frac{3}{4} & - \frac{3}{8} & - \frac{1}{4} \end{array}]$

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