Matemática discreta Ejemplos

$[\begin{array}{c} 4000 & 300 & 10 & 4319 \\ 3000 & 0 & 0 & 3003 \\ 2000 & 900 & 90 & 2996 \\ 1000 & 0 & 10 & 1015 \end{array}]$

Paso 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 $2$ by its cofactor and add.

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Paso 1.1

Consider the corresponding sign chart.

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

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 $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 300 & 10 & 4319 \\ 900 & 90 & 2996 \\ 0 & 10 & 1015 \end{array} |$

Paso 1.4

Multiply element $a_{21}$ by its cofactor.

$- 3000 | \begin{array}{c} 300 & 10 & 4319 \\ 900 & 90 & 2996 \\ 0 & 10 & 1015 \end{array} |$

Paso 1.5

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

$| \begin{array}{c} 4000 & 10 & 4319 \\ 2000 & 90 & 2996 \\ 1000 & 10 & 1015 \end{array} |$

Paso 1.6

Multiply element $a_{22}$ by its cofactor.

$0 | \begin{array}{c} 4000 & 10 & 4319 \\ 2000 & 90 & 2996 \\ 1000 & 10 & 1015 \end{array} |$

Paso 1.7

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

$| \begin{array}{c} 4000 & 300 & 4319 \\ 2000 & 900 & 2996 \\ 1000 & 0 & 1015 \end{array} |$

Paso 1.8

Multiply element $a_{23}$ by its cofactor.

$0 | \begin{array}{c} 4000 & 300 & 4319 \\ 2000 & 900 & 2996 \\ 1000 & 0 & 1015 \end{array} |$

Paso 1.9

The minor for $a_{24}$ is the determinant with row $2$ and column $4$ deleted.

$| \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 1.10

Multiply element $a_{24}$ by its cofactor.

$3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 1.11

Add the terms together.

$- 3000 | \begin{array}{c} 300 & 10 & 4319 \\ 900 & 90 & 2996 \\ 0 & 10 & 1015 \end{array} | + 0 | \begin{array}{c} 4000 & 10 & 4319 \\ 2000 & 90 & 2996 \\ 1000 & 10 & 1015 \end{array} | + 0 | \begin{array}{c} 4000 & 300 & 4319 \\ 2000 & 900 & 2996 \\ 1000 & 0 & 1015 \end{array} | + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 2

Multiplica $0$ por $| \begin{array}{c} 4000 & 10 & 4319 \\ 2000 & 90 & 2996 \\ 1000 & 10 & 1015 \end{array} |$ .

$- 3000 | \begin{array}{c} 300 & 10 & 4319 \\ 900 & 90 & 2996 \\ 0 & 10 & 1015 \end{array} | + 0 + 0 | \begin{array}{c} 4000 & 300 & 4319 \\ 2000 & 900 & 2996 \\ 1000 & 0 & 1015 \end{array} | + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 3

Multiplica $0$ por $| \begin{array}{c} 4000 & 300 & 4319 \\ 2000 & 900 & 2996 \\ 1000 & 0 & 1015 \end{array} |$ .

$- 3000 | \begin{array}{c} 300 & 10 & 4319 \\ 900 & 90 & 2996 \\ 0 & 10 & 1015 \end{array} | + 0 + 0 + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 4

Evalúa $| \begin{array}{c} 300 & 10 & 4319 \\ 900 & 90 & 2996 \\ 0 & 10 & 1015 \end{array} |$ .

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Paso 4.1

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 column $1$ by its cofactor and add.

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Paso 4.1.1

Consider the corresponding sign chart.

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

Paso 4.1.2

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

Paso 4.1.3

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

$| \begin{array}{c} 90 & 2996 \\ 10 & 1015 \end{array} |$

Paso 4.1.4

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

$300 | \begin{array}{c} 90 & 2996 \\ 10 & 1015 \end{array} |$

Paso 4.1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 10 & 4319 \\ 10 & 1015 \end{array} |$

Paso 4.1.6

Multiply element $a_{21}$ by its cofactor.

$- 900 | \begin{array}{c} 10 & 4319 \\ 10 & 1015 \end{array} |$

Paso 4.1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} 10 & 4319 \\ 90 & 2996 \end{array} |$

Paso 4.1.8

Multiply element $a_{31}$ by its cofactor.

$0 | \begin{array}{c} 10 & 4319 \\ 90 & 2996 \end{array} |$

Paso 4.1.9

Add the terms together.

$- 3000 (300 | \begin{array}{c} 90 & 2996 \\ 10 & 1015 \end{array} | - 900 | \begin{array}{c} 10 & 4319 \\ 10 & 1015 \end{array} | + 0 | \begin{array}{c} 10 & 4319 \\ 90 & 2996 \end{array} |) + 0 + 0 + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 4.2

Multiplica $0$ por $| \begin{array}{c} 10 & 4319 \\ 90 & 2996 \end{array} |$ .

$- 3000 (300 | \begin{array}{c} 90 & 2996 \\ 10 & 1015 \end{array} | - 900 | \begin{array}{c} 10 & 4319 \\ 10 & 1015 \end{array} | + 0) + 0 + 0 + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 4.3

Evalúa $| \begin{array}{c} 90 & 2996 \\ 10 & 1015 \end{array} |$ .

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Paso 4.3.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 3000 (300 (90 \cdot 1015 - 10 \cdot 2996) - 900 | \begin{array}{c} 10 & 4319 \\ 10 & 1015 \end{array} | + 0) + 0 + 0 + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 4.3.2

Simplifica el determinante.

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Paso 4.3.2.1

Simplifica cada término.

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Paso 4.3.2.1.1

Multiplica $90$ por $1015$ .

$- 3000 (300 (91350 - 10 \cdot 2996) - 900 | \begin{array}{c} 10 & 4319 \\ 10 & 1015 \end{array} | + 0) + 0 + 0 + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 4.3.2.1.2

Multiplica $- 10$ por $2996$ .

$- 3000 (300 (91350 - 29960) - 900 | \begin{array}{c} 10 & 4319 \\ 10 & 1015 \end{array} | + 0) + 0 + 0 + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 4.3.2.2

Resta $29960$ de $91350$ .

$- 3000 (300 \cdot 61390 - 900 | \begin{array}{c} 10 & 4319 \\ 10 & 1015 \end{array} | + 0) + 0 + 0 + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 4.4

Evalúa $| \begin{array}{c} 10 & 4319 \\ 10 & 1015 \end{array} |$ .

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Paso 4.4.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 3000 (300 \cdot 61390 - 900 (10 \cdot 1015 - 10 \cdot 4319) + 0) + 0 + 0 + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 4.4.2

Simplifica el determinante.

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Paso 4.4.2.1

Simplifica cada término.

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Paso 4.4.2.1.1

Multiplica $10$ por $1015$ .

$- 3000 (300 \cdot 61390 - 900 (10150 - 10 \cdot 4319) + 0) + 0 + 0 + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 4.4.2.1.2

Multiplica $- 10$ por $4319$ .

$- 3000 (300 \cdot 61390 - 900 (10150 - 43190) + 0) + 0 + 0 + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 4.4.2.2

Resta $43190$ de $10150$ .

$- 3000 (300 \cdot 61390 - 900 \cdot - 33040 + 0) + 0 + 0 + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 4.5

Simplifica el determinante.

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Paso 4.5.1

Simplifica cada término.

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Paso 4.5.1.1

Multiplica $300$ por $61390$ .

$- 3000 (18417000 - 900 \cdot - 33040 + 0) + 0 + 0 + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 4.5.1.2

Multiplica $- 900$ por $- 33040$ .

$- 3000 (18417000 + 29736000 + 0) + 0 + 0 + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 4.5.2

Suma $18417000$ y $29736000$ .

$- 3000 (48153000 + 0) + 0 + 0 + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 4.5.3

Suma $48153000$ y $0$ .

$- 3000 \cdot 48153000 + 0 + 0 + 3003 | \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$

Paso 5

Evalúa $| \begin{array}{c} 4000 & 300 & 10 \\ 2000 & 900 & 90 \\ 1000 & 0 & 10 \end{array} |$ .

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Paso 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 column $2$ by its cofactor and add.

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Paso 5.1.1

Consider the corresponding sign chart.

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

Paso 5.1.2

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

Paso 5.1.3

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

$| \begin{array}{c} 2000 & 90 \\ 1000 & 10 \end{array} |$

Paso 5.1.4

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

$- 300 | \begin{array}{c} 2000 & 90 \\ 1000 & 10 \end{array} |$

Paso 5.1.5

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

$| \begin{array}{c} 4000 & 10 \\ 1000 & 10 \end{array} |$

Paso 5.1.6

Multiply element $a_{22}$ by its cofactor.

$900 | \begin{array}{c} 4000 & 10 \\ 1000 & 10 \end{array} |$

Paso 5.1.7

The minor for $a_{32}$ is the determinant with row $3$ and column $2$ deleted.

$| \begin{array}{c} 4000 & 10 \\ 2000 & 90 \end{array} |$

Paso 5.1.8

Multiply element $a_{32}$ by its cofactor.

$0 | \begin{array}{c} 4000 & 10 \\ 2000 & 90 \end{array} |$

Paso 5.1.9

Add the terms together.

$- 3000 \cdot 48153000 + 0 + 0 + 3003 (- 300 | \begin{array}{c} 2000 & 90 \\ 1000 & 10 \end{array} | + 900 | \begin{array}{c} 4000 & 10 \\ 1000 & 10 \end{array} | + 0 | \begin{array}{c} 4000 & 10 \\ 2000 & 90 \end{array} |)$

Paso 5.2

Multiplica $0$ por $| \begin{array}{c} 4000 & 10 \\ 2000 & 90 \end{array} |$ .

$- 3000 \cdot 48153000 + 0 + 0 + 3003 (- 300 | \begin{array}{c} 2000 & 90 \\ 1000 & 10 \end{array} | + 900 | \begin{array}{c} 4000 & 10 \\ 1000 & 10 \end{array} | + 0)$

Paso 5.3

Evalúa $| \begin{array}{c} 2000 & 90 \\ 1000 & 10 \end{array} |$ .

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Paso 5.3.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 3000 \cdot 48153000 + 0 + 0 + 3003 (- 300 (2000 \cdot 10 - 1000 \cdot 90) + 900 | \begin{array}{c} 4000 & 10 \\ 1000 & 10 \end{array} | + 0)$

Paso 5.3.2

Simplifica el determinante.

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Paso 5.3.2.1

Simplifica cada término.

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Paso 5.3.2.1.1

Multiplica $2000$ por $10$ .

$- 3000 \cdot 48153000 + 0 + 0 + 3003 (- 300 (20000 - 1000 \cdot 90) + 900 | \begin{array}{c} 4000 & 10 \\ 1000 & 10 \end{array} | + 0)$

Paso 5.3.2.1.2

Multiplica $- 1000$ por $90$ .

$- 3000 \cdot 48153000 + 0 + 0 + 3003 (- 300 (20000 - 90000) + 900 | \begin{array}{c} 4000 & 10 \\ 1000 & 10 \end{array} | + 0)$

Paso 5.3.2.2

Resta $90000$ de $20000$ .

$- 3000 \cdot 48153000 + 0 + 0 + 3003 (- 300 \cdot - 70000 + 900 | \begin{array}{c} 4000 & 10 \\ 1000 & 10 \end{array} | + 0)$

Paso 5.4

Evalúa $| \begin{array}{c} 4000 & 10 \\ 1000 & 10 \end{array} |$ .

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Paso 5.4.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 3000 \cdot 48153000 + 0 + 0 + 3003 (- 300 \cdot - 70000 + 900 (4000 \cdot 10 - 1000 \cdot 10) + 0)$

Paso 5.4.2

Simplifica el determinante.

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Paso 5.4.2.1

Simplifica cada término.

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Paso 5.4.2.1.1

Multiplica $4000$ por $10$ .

$- 3000 \cdot 48153000 + 0 + 0 + 3003 (- 300 \cdot - 70000 + 900 (40000 - 1000 \cdot 10) + 0)$

Paso 5.4.2.1.2

Multiplica $- 1000$ por $10$ .

$- 3000 \cdot 48153000 + 0 + 0 + 3003 (- 300 \cdot - 70000 + 900 (40000 - 10000) + 0)$

Paso 5.4.2.2

Resta $10000$ de $40000$ .

$- 3000 \cdot 48153000 + 0 + 0 + 3003 (- 300 \cdot - 70000 + 900 \cdot 30000 + 0)$

Paso 5.5

Simplifica el determinante.

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Paso 5.5.1

Simplifica cada término.

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Paso 5.5.1.1

Multiplica $- 300$ por $- 70000$ .

$- 3000 \cdot 48153000 + 0 + 0 + 3003 (21000000 + 900 \cdot 30000 + 0)$

Paso 5.5.1.2

Multiplica $900$ por $30000$ .

$- 3000 \cdot 48153000 + 0 + 0 + 3003 (21000000 + 27000000 + 0)$

Paso 5.5.2

Suma $21000000$ y $27000000$ .

$- 3000 \cdot 48153000 + 0 + 0 + 3003 (48000000 + 0)$

Paso 5.5.3

Suma $48000000$ y $0$ .

$- 3000 \cdot 48153000 + 0 + 0 + 3003 \cdot 48000000$

Paso 6

Simplifica el determinante.

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Paso 6.1

Simplifica cada término.

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Paso 6.1.1

Multiplica $- 3000$ por $48153000$ .

$- 144459000000 + 0 + 0 + 3003 \cdot 48000000$

Paso 6.1.2

Multiplica $3003$ por $48000000$ .

$- 144459000000 + 0 + 0 + 144144000000$

Paso 6.2

Suma $- 144459000000$ y $0$ .

$- 144459000000 + 0 + 144144000000$

Paso 6.3

Suma $- 144459000000$ y $0$ .

$- 144459000000 + 144144000000$

Paso 6.4

Suma $- 144459000000$ y $144144000000$ .

$- 315000000$