Álgebra lineal Ejemplos

$[\begin{array}{c} a & b & 0 & 0 \\ c & d & 0 & 0 \\ 0 & 0 & a & - b \\ 0 & 0 & c & d \end{array}]$

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

$| \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$

Paso 1.4

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

$a | \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$

Paso 1.5

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

$| \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$

Paso 1.6

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

$- b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$

Paso 1.7

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

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

Paso 1.8

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

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

Paso 1.9

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

$| \begin{array}{c} c & d & 0 \\ 0 & 0 & a \\ 0 & 0 & c \end{array} |$

Paso 1.10

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

$0 | \begin{array}{c} c & d & 0 \\ 0 & 0 & a \\ 0 & 0 & c \end{array} |$

Paso 1.11

Add the terms together.

$a | \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 | \begin{array}{c} c & d & 0 \\ 0 & 0 & - b \\ 0 & 0 & d \end{array} | + 0 | \begin{array}{c} c & d & 0 \\ 0 & 0 & a \\ 0 & 0 & c \end{array} |$

Paso 2

Multiplica $0$ por $| \begin{array}{c} c & d & 0 \\ 0 & 0 & - b \\ 0 & 0 & d \end{array} |$ .

$a | \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0 | \begin{array}{c} c & d & 0 \\ 0 & 0 & a \\ 0 & 0 & c \end{array} |$

Paso 3

Multiplica $0$ por $| \begin{array}{c} c & d & 0 \\ 0 & 0 & a \\ 0 & 0 & c \end{array} |$ .

$a | \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Paso 4

Evalúa $| \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$ .

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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 row $1$ 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.

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

Paso 4.1.4

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

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

Paso 4.1.5

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

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

Paso 4.1.6

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

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

Paso 4.1.7

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

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

Paso 4.1.8

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

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

Paso 4.1.9

Add the terms together.

$a (d | \begin{array}{c} a & - b \\ c & d \end{array} | + 0 | \begin{array}{c} 0 & - b \\ 0 & d \end{array} | + 0 | \begin{array}{c} 0 & a \\ 0 & c \end{array} |) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Paso 4.2

Multiplica $0$ por $| \begin{array}{c} 0 & - b \\ 0 & d \end{array} |$ .

$a (d | \begin{array}{c} a & - b \\ c & d \end{array} | + 0 + 0 | \begin{array}{c} 0 & a \\ 0 & c \end{array} |) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Paso 4.3

Multiplica $0$ por $| \begin{array}{c} 0 & a \\ 0 & c \end{array} |$ .

$a (d | \begin{array}{c} a & - b \\ c & d \end{array} | + 0 + 0) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Paso 4.4

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

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Paso 4.4.1

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

$a (d (a d - c (- b)) + 0 + 0) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Paso 4.4.2

Simplifica cada término.

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Paso 4.4.2.1

Reescribe con la propiedad conmutativa de la multiplicación.

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

Paso 4.4.2.2

Multiplica $- 1$ por $- 1$ .

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

Paso 4.4.2.3

Multiplica $c$ por $1$ .

$a (d (a d + c b) + 0 + 0) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Paso 4.5

Simplifica el determinante.

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Paso 4.5.1

Combina los términos opuestos en $d (a d + c b) + 0 + 0$ .

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Paso 4.5.1.1

Suma $d (a d + c b)$ y $0$ .

$a (d (a d + c b) + 0) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Paso 4.5.1.2

Suma $d (a d + c b)$ y $0$ .

$a (d (a d + c b)) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Paso 4.5.2

Aplica la propiedad distributiva.

$a (d (a d) + d (c b)) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Paso 4.5.3

Multiplica $d$ por $d$ sumando los exponentes.

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Paso 4.5.3.1

Mueve $d$ .

$a (d \cdot d a + d (c b)) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Paso 4.5.3.2

Multiplica $d$ por $d$ .

$a (d^{2} a + d (c b)) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

$a (d^{2} a + d c b) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Paso 5

Evalúa $| \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$ .

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

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

Paso 5.1.4

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

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

Paso 5.1.5

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

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

Paso 5.1.6

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

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

Paso 5.1.7

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

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

Paso 5.1.8

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

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

Paso 5.1.9

Add the terms together.

$a (d^{2} a + d c b) - b (c | \begin{array}{c} a & - b \\ c & d \end{array} | + 0 | \begin{array}{c} 0 & - b \\ 0 & d \end{array} | + 0 | \begin{array}{c} 0 & a \\ 0 & c \end{array} |) + 0 + 0$

Paso 5.2

Multiplica $0$ por $| \begin{array}{c} 0 & - b \\ 0 & d \end{array} |$ .

$a (d^{2} a + d c b) - b (c | \begin{array}{c} a & - b \\ c & d \end{array} | + 0 + 0 | \begin{array}{c} 0 & a \\ 0 & c \end{array} |) + 0 + 0$

Paso 5.3

Multiplica $0$ por $| \begin{array}{c} 0 & a \\ 0 & c \end{array} |$ .

$a (d^{2} a + d c b) - b (c | \begin{array}{c} a & - b \\ c & d \end{array} | + 0 + 0) + 0 + 0$

Paso 5.4

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

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Paso 5.4.1

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

$a (d^{2} a + d c b) - b (c (a d - c (- b)) + 0 + 0) + 0 + 0$

Paso 5.4.2

Simplifica cada término.

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Paso 5.4.2.1

Reescribe con la propiedad conmutativa de la multiplicación.

$a (d^{2} a + d c b) - b (c (a d - 1 \cdot - 1 c b) + 0 + 0) + 0 + 0$

Paso 5.4.2.2

Multiplica $- 1$ por $- 1$ .

$a (d^{2} a + d c b) - b (c (a d + 1 c b) + 0 + 0) + 0 + 0$

Paso 5.4.2.3

Multiplica $c$ por $1$ .

$a (d^{2} a + d c b) - b (c (a d + c b) + 0 + 0) + 0 + 0$

Paso 5.5

Simplifica el determinante.

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Paso 5.5.1

Combina los términos opuestos en $c (a d + c b) + 0 + 0$ .

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Paso 5.5.1.1

Suma $c (a d + c b)$ y $0$ .

$a (d^{2} a + d c b) - b (c (a d + c b) + 0) + 0 + 0$

Paso 5.5.1.2

Suma $c (a d + c b)$ y $0$ .

$a (d^{2} a + d c b) - b (c (a d + c b)) + 0 + 0$

Paso 5.5.2

Aplica la propiedad distributiva.

$a (d^{2} a + d c b) - b (c (a d) + c (c b)) + 0 + 0$

Paso 5.5.3

Multiplica $c$ por $c$ sumando los exponentes.

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Paso 5.5.3.1

Mueve $c$ .

$a (d^{2} a + d c b) - b (c a d + c \cdot c b) + 0 + 0$

Paso 5.5.3.2

Multiplica $c$ por $c$ .

$a (d^{2} a + d c b) - b (c a d + c^{2} b) + 0 + 0$

Paso 6

Simplifica el determinante.

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Paso 6.1

Combina los términos opuestos en $a (d^{2} a + d c b) - b (c a d + c^{2} b) + 0 + 0$ .

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Paso 6.1.1

Suma $a (d^{2} a + d c b) - b (c a d + c^{2} b)$ y $0$ .

$a (d^{2} a + d c b) - b (c a d + c^{2} b) + 0$

Paso 6.1.2

Suma $a (d^{2} a + d c b) - b (c a d + c^{2} b)$ y $0$ .

$a (d^{2} a + d c b) - b (c a d + c^{2} b)$

Paso 6.2

Simplifica cada término.

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Paso 6.2.1

Aplica la propiedad distributiva.

$a (d^{2} a) + a (d c b) - b (c a d + c^{2} b)$

Paso 6.2.2

Multiplica $a$ por $a$ sumando los exponentes.

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Paso 6.2.2.1

Mueve $a$ .

$a \cdot a d^{2} + a (d c b) - b (c a d + c^{2} b)$

Paso 6.2.2.2

Multiplica $a$ por $a$ .

$a^{2} d^{2} + a (d c b) - b (c a d + c^{2} b)$

Paso 6.2.3

Aplica la propiedad distributiva.

$a^{2} d^{2} + a d c b - b (c a d) - b (c^{2} b)$

Paso 6.2.4

Multiplica $b$ por $b$ sumando los exponentes.

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Paso 6.2.4.1

Mueve $b$ .

$a^{2} d^{2} + a d c b - b c a d - (b \cdot b) c^{2}$

Paso 6.2.4.2

Multiplica $b$ por $b$ .

$a^{2} d^{2} + a d c b - b c a d - b^{2} c^{2}$

Paso 6.3

Combina los términos opuestos en $a^{2} d^{2} + a d c b - b c a d - b^{2} c^{2}$ .

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Paso 6.3.1

Reordena los factores en los términos $a d c b$ y $- b c a d$ .

$a^{2} d^{2} + a b c d - a b c d - b^{2} c^{2}$

Paso 6.3.2

Resta $a b c d$ de $a b c d$ .

$a^{2} d^{2} + 0 - b^{2} c^{2}$

Paso 6.3.3

Suma $a^{2} d^{2}$ y $0$ .

$a^{2} d^{2} - b^{2} c^{2}$