Matemática discreta Ejemplos

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

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

Paso 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

3

$3$ by its cofactor and add.

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Paso 1.1

Consider the corresponding sign chart.

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

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

Paso 1.2

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

-

$-$ position on the sign chart.

Paso 1.3

The minor for

a_{13}

$a_{13}$ is the determinant with row

1

$1$ and column

3

$3$ deleted.

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

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

Paso 1.4

Multiply element

a_{13}

$a_{13}$ by its cofactor.

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

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

Paso 1.5

The minor for

a_{23}

$a_{23}$ is the determinant with row

2

$2$ and column

3

$3$ deleted.

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

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

Paso 1.6

Multiply element

a_{23}

$a_{23}$ by its cofactor.

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

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

Paso 1.7

The minor for

a_{33}

$a_{33}$ is the determinant with row

3

$3$ and column

3

$3$ deleted.

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

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

Paso 1.8

Multiply element

a_{33}

$a_{33}$ by its cofactor.

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

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

Paso 1.9

Add the terms together.

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

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

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

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

Paso 2

Multiplica

0

$0$ por

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

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

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

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

Paso 3

Multiplica

0

$0$ por

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

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

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

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

Paso 4

Evalúa

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

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

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Paso 4.1

El determinante de una matriz

2 \times 2

$2 \times 2$ puede obtenerse usando la fórmula

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

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

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

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

Paso 4.2

Simplifica el determinante.

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Paso 4.2.1

Simplifica cada término.

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Paso 4.2.1.1

Multiplica

1

$1$ por

1

$1$ .

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

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

Paso 4.2.1.2

Multiplica

- 1

$- 1$ por

1

$1$ .

2 (1 - 1) + 0 + 0

$2 (1 - 1) + 0 + 0$

2 (1 - 1) + 0 + 0

$2 (1 - 1) + 0 + 0$

Paso 4.2.2

Resta

1

$1$ de

1

$1$ .

2 \cdot 0 + 0 + 0

$2 \cdot 0 + 0 + 0$

2 \cdot 0 + 0 + 0

$2 \cdot 0 + 0 + 0$

2 \cdot 0 + 0 + 0

$2 \cdot 0 + 0 + 0$

Paso 5

Simplifica el determinante.

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Paso 5.1

Multiplica

2

$2$ por

0

$0$ .

0 + 0 + 0

$0 + 0 + 0$

Paso 5.2

Suma

0

$0$ y

0

$0$ .

0 + 0

$0 + 0$

Paso 5.3

Suma

0

$0$ y

0

$0$ .

0

$0$

0

$0$

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