Álgebra lineal Ejemplos

$[\begin{array}{c} x & y & z \\ x^{2} & y^{2} & z^{2} \\ x^{3} & y^{3} & z^{3} \end{array}]$

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.

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

Paso 1.4

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

$x | \begin{array}{c} y^{2} & z^{2} \\ y^{3} & z^{3} \end{array} |$

Paso 1.5

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

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

Paso 1.6

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

$- y | \begin{array}{c} x^{2} & z^{2} \\ x^{3} & z^{3} \end{array} |$

Paso 1.7

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

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

Paso 1.8

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

$z | \begin{array}{c} x^{2} & y^{2} \\ x^{3} & y^{3} \end{array} |$

Paso 1.9

Add the terms together.

$x | \begin{array}{c} y^{2} & z^{2} \\ y^{3} & z^{3} \end{array} | - y | \begin{array}{c} x^{2} & z^{2} \\ x^{3} & z^{3} \end{array} | + z | \begin{array}{c} x^{2} & y^{2} \\ x^{3} & y^{3} \end{array} |$

Paso 2

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

$x (y^{2} z^{3} - y^{3} z^{2}) - y | \begin{array}{c} x^{2} & z^{2} \\ x^{3} & z^{3} \end{array} | + z | \begin{array}{c} x^{2} & y^{2} \\ x^{3} & y^{3} \end{array} |$

Paso 3

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

$x (y^{2} z^{3} - y^{3} z^{2}) - y (x^{2} z^{3} - x^{3} z^{2}) + z | \begin{array}{c} x^{2} & y^{2} \\ x^{3} & y^{3} \end{array} |$

Paso 4

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

$x (y^{2} z^{3} - y^{3} z^{2}) - y (x^{2} z^{3} - x^{3} z^{2}) + z (x^{2} y^{3} - x^{3} y^{2})$

Paso 5

Simplifica cada término.

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Paso 5.1

Aplica la propiedad distributiva.

$x (y^{2} z^{3}) + x (- y^{3} z^{2}) - y (x^{2} z^{3} - x^{3} z^{2}) + z (x^{2} y^{3} - x^{3} y^{2})$

Paso 5.2

Reescribe con la propiedad conmutativa de la multiplicación.

$x y^{2} z^{3} - x y^{3} z^{2} - y (x^{2} z^{3} - x^{3} z^{2}) + z (x^{2} y^{3} - x^{3} y^{2})$

Paso 5.3

Aplica la propiedad distributiva.

$x y^{2} z^{3} - x y^{3} z^{2} - y (x^{2} z^{3}) - y (- x^{3} z^{2}) + z (x^{2} y^{3} - x^{3} y^{2})$

Paso 5.4

Multiplica $- y (- x^{3} z^{2})$ .

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Paso 5.4.1

Multiplica $- 1$ por $- 1$ .

$x y^{2} z^{3} - x y^{3} z^{2} - y x^{2} z^{3} + 1 y (x^{3} z^{2}) + z (x^{2} y^{3} - x^{3} y^{2})$

Paso 5.4.2

Multiplica $y$ por $1$ .

$x y^{2} z^{3} - x y^{3} z^{2} - y x^{2} z^{3} + y (x^{3} z^{2}) + z (x^{2} y^{3} - x^{3} y^{2})$

Paso 5.5

Aplica la propiedad distributiva.

$x y^{2} z^{3} - x y^{3} z^{2} - y x^{2} z^{3} + y x^{3} z^{2} + z (x^{2} y^{3}) + z (- x^{3} y^{2})$

Paso 5.6

Reescribe con la propiedad conmutativa de la multiplicación.

$x y^{2} z^{3} - x y^{3} z^{2} - y x^{2} z^{3} + y x^{3} z^{2} + z x^{2} y^{3} - z x^{3} y^{2}$