Matematica discreta Esempi

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

Passaggio 1

Find the determinant.

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Passaggio 1.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Passaggio 1.1.1

Consider the corresponding sign chart.

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

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 $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

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

Passaggio 1.1.4

Multiply element $a_{11}$ by its cofactor.

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

Passaggio 1.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

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

Passaggio 1.1.6

Multiply element $a_{12}$ by its cofactor.

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

Passaggio 1.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

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

Passaggio 1.1.8

Multiply element $a_{13}$ by its cofactor.

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

Passaggio 1.1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

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

Passaggio 1.1.10

Multiply element $a_{14}$ by its cofactor.

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

Passaggio 1.1.11

Add the terms together.

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

Passaggio 1.2

Moltiplica $0$ per $| \begin{array}{c} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array} |$ .

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

Passaggio 1.3

Moltiplica $0$ per $| \begin{array}{c} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} |$ .

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

Passaggio 1.4

Moltiplica $0$ per $| \begin{array}{c} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array} |$ .

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

Passaggio 1.5

Calcola $| \begin{array}{c} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} |$ .

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Passaggio 1.5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Passaggio 1.5.1.1

Consider the corresponding sign chart.

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

Passaggio 1.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Passaggio 1.5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

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

Passaggio 1.5.1.4

Multiply element $a_{11}$ by its cofactor.

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

Passaggio 1.5.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

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

Passaggio 1.5.1.6

Multiply element $a_{12}$ by its cofactor.

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

Passaggio 1.5.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

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

Passaggio 1.5.1.8

Multiply element $a_{13}$ by its cofactor.

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

Passaggio 1.5.1.9

Add the terms together.

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

Passaggio 1.5.2

Moltiplica $0$ per $| \begin{array}{c} 0 & 0 \\ 0 & 1 \end{array} |$ .

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

Passaggio 1.5.3

Moltiplica $0$ per $| \begin{array}{c} 1 & 0 \\ 0 & 0 \end{array} |$ .

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

Passaggio 1.5.4

Calcola $| \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} |$ .

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Passaggio 1.5.4.1

È possibile trovare il determinante di una matrice $2 \times 2$ usando la formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$0 + 0 + 1 (0 - 1 (1 \cdot 1 + 0 \cdot 0) + 0) + 0$

Passaggio 1.5.4.2

Semplifica il determinante.

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Passaggio 1.5.4.2.1

Semplifica ciascun termine.

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Passaggio 1.5.4.2.1.1

Moltiplica $1$ per $1$ .

$0 + 0 + 1 (0 - 1 (1 + 0 \cdot 0) + 0) + 0$

Passaggio 1.5.4.2.1.2

Moltiplica $0$ per $0$ .

$0 + 0 + 1 (0 - 1 (1 + 0) + 0) + 0$

Passaggio 1.5.4.2.2

Somma $1$ e $0$ .

$0 + 0 + 1 (0 - 1 \cdot 1 + 0) + 0$

Passaggio 1.5.5

Semplifica il determinante.

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Passaggio 1.5.5.1

Moltiplica $- 1$ per $1$ .

$0 + 0 + 1 (0 - 1 + 0) + 0$

Passaggio 1.5.5.2

Sottrai $1$ da $0$ .

$0 + 0 + 1 (- 1 + 0) + 0$

Passaggio 1.5.5.3

Somma $- 1$ e $0$ .

$0 + 0 + 1 \cdot - 1 + 0$

Passaggio 1.6

Semplifica il determinante.

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Passaggio 1.6.1

Moltiplica $- 1$ per $1$ .

$0 + 0 - 1 + 0$

Passaggio 1.6.2

Somma $0$ e $0$ .

$0 - 1 + 0$

Passaggio 1.6.3

Sottrai $1$ da $0$ .

$- 1 + 0$

Passaggio 1.6.4

Somma $- 1$ e $0$ .

$- 1$

Passaggio 2

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

Passaggio 3

Set up a $4 \times 8$ matrix where the left half is the original matrix and the right half is its identity matrix.

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

Passaggio 4

Trova la forma ridotta a scala per righe di Echelon.

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Passaggio 4.1

Swap $R_{3}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

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

Passaggio 5

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

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