Álgebra lineal Ejemplos

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 4 & 5 & 0 0 & 2 & 1 & 3 2 & 5 & 4 & 1 1 & 5 & 0 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 4 & 5 & 0 \\ 0 & 2 & 1 & 3 \\ 2 & 5 & 4 & 1 \\ 1 & 5 & 0 & 2 \end{array}]$

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 row

1

$1$ by its cofactor and add.

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Paso 1.1

Consider the corresponding sign chart.

∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} + & - & + & - - & + & - & + + & - & + & - - & + & - & + \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣

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

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}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

∣ ∣ ∣ ∣ \begin{matrix} 2 & 1 & 3 5 & 4 & 1 5 & 0 & 2 \end{matrix} ∣ ∣ ∣ ∣

$| \begin{array}{c} 2 & 1 & 3 \\ 5 & 4 & 1 \\ 5 & 0 & 2 \end{array} |$

Paso 1.4

Multiply element

a_{11}

$a_{11}$ by its cofactor.

1 ∣ ∣ ∣ ∣ \begin{matrix} 2 & 1 & 3 5 & 4 & 1 5 & 0 & 2 \end{matrix} ∣ ∣ ∣ ∣

$1 | \begin{array}{c} 2 & 1 & 3 \\ 5 & 4 & 1 \\ 5 & 0 & 2 \end{array} |$

Paso 1.5

The minor for

a_{12}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

∣ ∣ ∣ ∣ \begin{matrix} 0 & 1 & 3 2 & 4 & 1 1 & 0 & 2 \end{matrix} ∣ ∣ ∣ ∣

$| \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} |$

Paso 1.6

Multiply element

a_{12}

$a_{12}$ by its cofactor.

- 4 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 1 & 3 2 & 4 & 1 1 & 0 & 2 \end{matrix} ∣ ∣ ∣ ∣

$- 4 | \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} |$

Paso 1.7

The minor for

a_{13}

$a_{13}$ is the determinant with row

1

$1$ and column

3

$3$ deleted.

∣ ∣ ∣ ∣ \begin{matrix} 0 & 2 & 3 2 & 5 & 1 1 & 5 & 2 \end{matrix} ∣ ∣ ∣ ∣

$| \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} |$

Paso 1.8

Multiply element

a_{13}

$a_{13}$ by its cofactor.

5 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 2 & 3 2 & 5 & 1 1 & 5 & 2 \end{matrix} ∣ ∣ ∣ ∣

$5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} |$

Paso 1.9

The minor for

a_{14}

$a_{14}$ is the determinant with row

1

$1$ and column

4

$4$ deleted.

∣ ∣ ∣ ∣ \begin{matrix} 0 & 2 & 1 2 & 5 & 4 1 & 5 & 0 \end{matrix} ∣ ∣ ∣ ∣

$| \begin{array}{c} 0 & 2 & 1 \\ 2 & 5 & 4 \\ 1 & 5 & 0 \end{array} |$

Paso 1.10

Multiply element

a_{14}

$a_{14}$ by its cofactor.

0 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 2 & 1 2 & 5 & 4 1 & 5 & 0 \end{matrix} ∣ ∣ ∣ ∣

$0 | \begin{array}{c} 0 & 2 & 1 \\ 2 & 5 & 4 \\ 1 & 5 & 0 \end{array} |$

Paso 1.11

Add the terms together.

1 ∣ ∣ ∣ ∣ \begin{matrix} 2 & 1 & 3 5 & 4 & 1 5 & 0 & 2 \end{matrix} ∣ ∣ ∣ ∣ - 4 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 1 & 3 2 & 4 & 1 1 & 0 & 2 \end{matrix} ∣ ∣ ∣ ∣ + 5 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 2 & 3 2 & 5 & 1 1 & 5 & 2 \end{matrix} ∣ ∣ ∣ ∣ + 0 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 2 & 1 2 & 5 & 4 1 & 5 & 0 \end{matrix} ∣ ∣ ∣ ∣

$1 | \begin{array}{c} 2 & 1 & 3 \\ 5 & 4 & 1 \\ 5 & 0 & 2 \end{array} | - 4 | \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} | + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0 | \begin{array}{c} 0 & 2 & 1 \\ 2 & 5 & 4 \\ 1 & 5 & 0 \end{array} |$

1 ∣ ∣ ∣ ∣ \begin{matrix} 2 & 1 & 3 5 & 4 & 1 5 & 0 & 2 \end{matrix} ∣ ∣ ∣ ∣ - 4 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 1 & 3 2 & 4 & 1 1 & 0 & 2 \end{matrix} ∣ ∣ ∣ ∣ + 5 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 2 & 3 2 & 5 & 1 1 & 5 & 2 \end{matrix} ∣ ∣ ∣ ∣ + 0 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 2 & 1 2 & 5 & 4 1 & 5 & 0 \end{matrix} ∣ ∣ ∣ ∣

Paso 2

Multiplica

0

$0$ por

∣ ∣ ∣ ∣ \begin{matrix} 0 & 2 & 1 2 & 5 & 4 1 & 5 & 0 \end{matrix} ∣ ∣ ∣ ∣

$| \begin{array}{c} 0 & 2 & 1 \\ 2 & 5 & 4 \\ 1 & 5 & 0 \end{array} |$ .

1 ∣ ∣ ∣ ∣ \begin{matrix} 2 & 1 & 3 5 & 4 & 1 5 & 0 & 2 \end{matrix} ∣ ∣ ∣ ∣ - 4 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 1 & 3 2 & 4 & 1 1 & 0 & 2 \end{matrix} ∣ ∣ ∣ ∣ + 5 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 2 & 3 2 & 5 & 1 1 & 5 & 2 \end{matrix} ∣ ∣ ∣ ∣ + 0

Paso 3

Evalúa

∣ ∣ ∣ ∣ \begin{matrix} 2 & 1 & 3 5 & 4 & 1 5 & 0 & 2 \end{matrix} ∣ ∣ ∣ ∣

$| \begin{array}{c} 2 & 1 & 3 \\ 5 & 4 & 1 \\ 5 & 0 & 2 \end{array} |$ .

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Paso 3.1

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

2

$2$ by its cofactor and add.

Toca para ver más pasos...

Paso 3.1.1

Consider the corresponding sign chart.

∣ ∣ ∣ ∣ \begin{matrix} + & - & + - & + & - + & - & + \end{matrix} ∣ ∣ ∣ ∣

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

Paso 3.1.2

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

-

$-$ position on the sign chart.

Paso 3.1.3

The minor for

a_{12}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

∣ ∣ ∣ \begin{matrix} 5 & 1 5 & 2 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 5 & 1 \\ 5 & 2 \end{array} |$

Paso 3.1.4

Multiply element

a_{12}

$a_{12}$ by its cofactor.

- 1 ∣ ∣ ∣ \begin{matrix} 5 & 1 5 & 2 \end{matrix} ∣ ∣ ∣

$- 1 | \begin{array}{c} 5 & 1 \\ 5 & 2 \end{array} |$

Paso 3.1.5

The minor for

a_{22}

$a_{22}$ is the determinant with row

2

$2$ and column

2

$2$ deleted.

∣ ∣ ∣ \begin{matrix} 2 & 3 5 & 2 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 2 & 3 \\ 5 & 2 \end{array} |$

Paso 3.1.6

Multiply element

a_{22}

$a_{22}$ by its cofactor.

4 ∣ ∣ ∣ \begin{matrix} 2 & 3 5 & 2 \end{matrix} ∣ ∣ ∣

$4 | \begin{array}{c} 2 & 3 \\ 5 & 2 \end{array} |$

Paso 3.1.7

The minor for

a_{32}

$a_{32}$ is the determinant with row

3

$3$ and column

2

$2$ deleted.

∣ ∣ ∣ \begin{matrix} 2 & 3 5 & 1 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 2 & 3 \\ 5 & 1 \end{array} |$

Paso 3.1.8

Multiply element

a_{32}

$a_{32}$ by its cofactor.

0 ∣ ∣ ∣ \begin{matrix} 2 & 3 5 & 1 \end{matrix} ∣ ∣ ∣

$0 | \begin{array}{c} 2 & 3 \\ 5 & 1 \end{array} |$

Paso 3.1.9

Add the terms together.

1 (- 1 ∣ ∣ ∣ \begin{matrix} 5 & 1 5 & 2 \end{matrix} ∣ ∣ ∣ + 4 ∣ ∣ ∣ \begin{matrix} 2 & 3 5 & 2 \end{matrix} ∣ ∣ ∣ + 0 ∣ ∣ ∣ \begin{matrix} 2 & 3 5 & 1 \end{matrix} ∣ ∣ ∣) - 4 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 1 & 3 2 & 4 & 1 1 & 0 & 2 \end{matrix} ∣ ∣ ∣ ∣ + 5 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 2 & 3 2 & 5 & 1 1 & 5 & 2 \end{matrix} ∣ ∣ ∣ ∣ + 0

$1 (- 1 | \begin{array}{c} 5 & 1 \\ 5 & 2 \end{array} | + 4 | \begin{array}{c} 2 & 3 \\ 5 & 2 \end{array} | + 0 | \begin{array}{c} 2 & 3 \\ 5 & 1 \end{array} |) - 4 | \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} | + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

1 (- 1 ∣ ∣ ∣ \begin{matrix} 5 & 1 5 & 2 \end{matrix} ∣ ∣ ∣ + 4 ∣ ∣ ∣ \begin{matrix} 2 & 3 5 & 2 \end{matrix} ∣ ∣ ∣ + 0 ∣ ∣ ∣ \begin{matrix} 2 & 3 5 & 1 \end{matrix} ∣ ∣ ∣) - 4 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 1 & 3 2 & 4 & 1 1 & 0 & 2 \end{matrix} ∣ ∣ ∣ ∣ + 5 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 2 & 3 2 & 5 & 1 1 & 5 & 2 \end{matrix} ∣ ∣ ∣ ∣ + 0

Paso 3.2

Multiplica

0

$0$ por

∣ ∣ ∣ \begin{matrix} 2 & 3 5 & 1 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 2 & 3 \\ 5 & 1 \end{array} |$ .

1 (- 1 ∣ ∣ ∣ \begin{matrix} 5 & 1 5 & 2 \end{matrix} ∣ ∣ ∣ + 4 ∣ ∣ ∣ \begin{matrix} 2 & 3 5 & 2 \end{matrix} ∣ ∣ ∣ + 0) - 4 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 1 & 3 2 & 4 & 1 1 & 0 & 2 \end{matrix} ∣ ∣ ∣ ∣ + 5 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 2 & 3 2 & 5 & 1 1 & 5 & 2 \end{matrix} ∣ ∣ ∣ ∣ + 0

$1 (- 1 | \begin{array}{c} 5 & 1 \\ 5 & 2 \end{array} | + 4 | \begin{array}{c} 2 & 3 \\ 5 & 2 \end{array} | + 0) - 4 | \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} | + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 3.3

Evalúa

∣ ∣ ∣ \begin{matrix} 5 & 1 5 & 2 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 5 & 1 \\ 5 & 2 \end{array} |$ .

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Paso 3.3.1

El determinante de una matriz

2 \times 2

$2 \times 2$ puede obtenerse usando la fórmula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

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

1 (- 1 (5 \cdot 2 - 5 \cdot 1) + 4 ∣ ∣ ∣ \begin{matrix} 2 & 3 5 & 2 \end{matrix} ∣ ∣ ∣ + 0) - 4 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 1 & 3 2 & 4 & 1 1 & 0 & 2 \end{matrix} ∣ ∣ ∣ ∣ + 5 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 2 & 3 2 & 5 & 1 1 & 5 & 2 \end{matrix} ∣ ∣ ∣ ∣ + 0

$1 (- 1 (5 \cdot 2 - 5 \cdot 1) + 4 | \begin{array}{c} 2 & 3 \\ 5 & 2 \end{array} | + 0) - 4 | \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} | + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 3.3.2

Simplifica el determinante.

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Paso 3.3.2.1

Simplifica cada término.

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Paso 3.3.2.1.1

Multiplica

5

$5$ por

2

$2$ .

1 (- 1 (10 - 5 \cdot 1) + 4 ∣ ∣ ∣ \begin{matrix} 2 & 3 5 & 2 \end{matrix} ∣ ∣ ∣ + 0) - 4 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 1 & 3 2 & 4 & 1 1 & 0 & 2 \end{matrix} ∣ ∣ ∣ ∣ + 5 ∣ ∣ ∣ ∣ \begin{matrix} 0 & 2 & 3 2 & 5 & 1 1 & 5 & 2 \end{matrix} ∣ ∣ ∣ ∣ + 0

$1 (- 1 (10 - 5 \cdot 1) + 4 | \begin{array}{c} 2 & 3 \\ 5 & 2 \end{array} | + 0) - 4 | \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} | + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 3.3.2.1.2

Multiplica

$- 5$ por

$1$ .

$1 (- 1 (10 - 5) + 4 | \begin{array}{c} 2 & 3 \\ 5 & 2 \end{array} | + 0) - 4 | \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} | + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 3.3.2.2

Resta

$5$ de

$10$ .

$1 (- 1 \cdot 5 + 4 | \begin{array}{c} 2 & 3 \\ 5 & 2 \end{array} | + 0) - 4 | \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} | + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 3.4

Evalúa

$| \begin{array}{c} 2 & 3 \\ 5 & 2 \end{array} |$ .

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Paso 3.4.1

El determinante de una matriz

$2 \times 2$ puede obtenerse usando la fórmula

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

$1 (- 1 \cdot 5 + 4 (2 \cdot 2 - 5 \cdot 3) + 0) - 4 | \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} | + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 3.4.2

Simplifica el determinante.

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Paso 3.4.2.1

Simplifica cada término.

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Paso 3.4.2.1.1

Multiplica

$2$ por

$2$ .

$1 (- 1 \cdot 5 + 4 (4 - 5 \cdot 3) + 0) - 4 | \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} | + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 3.4.2.1.2

Multiplica

$- 5$ por

$3$ .

$1 (- 1 \cdot 5 + 4 (4 - 15) + 0) - 4 | \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} | + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 3.4.2.2

Resta

$15$ de

$4$ .

$1 (- 1 \cdot 5 + 4 \cdot - 11 + 0) - 4 | \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} | + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 3.5

Simplifica el determinante.

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Paso 3.5.1

Simplifica cada término.

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Paso 3.5.1.1

Multiplica

$- 1$ por

$5$ .

$1 (- 5 + 4 \cdot - 11 + 0) - 4 | \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} | + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 3.5.1.2

Multiplica

$4$ por

$- 11$ .

$1 (- 5 - 44 + 0) - 4 | \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} | + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 3.5.2

Resta

$44$ de

$- 5$ .

$1 (- 49 + 0) - 4 | \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} | + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 3.5.3

Suma

$- 49$ y

$0$ .

$1 \cdot - 49 - 4 | \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} | + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 4

Evalúa

$| \begin{array}{c} 0 & 1 & 3 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{array} |$ .

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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.

$| \begin{array}{c} 4 & 1 \\ 0 & 2 \end{array} |$

Paso 4.1.4

Multiply element

$a_{11}$ by its cofactor.

$0 | \begin{array}{c} 4 & 1 \\ 0 & 2 \end{array} |$

Paso 4.1.5

The minor for

$a_{12}$ is the determinant with row

$1$ and column

$2$ deleted.

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

Paso 4.1.6

Multiply element

$a_{12}$ by its cofactor.

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

Paso 4.1.7

The minor for

$a_{13}$ is the determinant with row

$1$ and column

$3$ deleted.

$| \begin{array}{c} 2 & 4 \\ 1 & 0 \end{array} |$

Paso 4.1.8

Multiply element

$a_{13}$ by its cofactor.

$3 | \begin{array}{c} 2 & 4 \\ 1 & 0 \end{array} |$

Paso 4.1.9

Add the terms together.

$1 \cdot - 49 - 4 (0 | \begin{array}{c} 4 & 1 \\ 0 & 2 \end{array} | - 1 | \begin{array}{c} 2 & 1 \\ 1 & 2 \end{array} | + 3 | \begin{array}{c} 2 & 4 \\ 1 & 0 \end{array} |) + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 4.2

Multiplica

$0$ por

$| \begin{array}{c} 4 & 1 \\ 0 & 2 \end{array} |$ .

$1 \cdot - 49 - 4 (0 - 1 | \begin{array}{c} 2 & 1 \\ 1 & 2 \end{array} | + 3 | \begin{array}{c} 2 & 4 \\ 1 & 0 \end{array} |) + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 4.3

Evalúa

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

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Paso 4.3.1

El determinante de una matriz

$2 \times 2$ puede obtenerse usando la fórmula

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

$1 \cdot - 49 - 4 (0 - 1 (2 \cdot 2 - 1 \cdot 1) + 3 | \begin{array}{c} 2 & 4 \\ 1 & 0 \end{array} |) + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

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

$2$ por

$2$ .

$1 \cdot - 49 - 4 (0 - 1 (4 - 1 \cdot 1) + 3 | \begin{array}{c} 2 & 4 \\ 1 & 0 \end{array} |) + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 4.3.2.1.2

Multiplica

$- 1$ por

$1$ .

$1 \cdot - 49 - 4 (0 - 1 (4 - 1) + 3 | \begin{array}{c} 2 & 4 \\ 1 & 0 \end{array} |) + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 4.3.2.2

Resta

$1$ de

$4$ .

$1 \cdot - 49 - 4 (0 - 1 \cdot 3 + 3 | \begin{array}{c} 2 & 4 \\ 1 & 0 \end{array} |) + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 4.4

Evalúa

$| \begin{array}{c} 2 & 4 \\ 1 & 0 \end{array} |$ .

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Paso 4.4.1

El determinante de una matriz

$2 \times 2$ puede obtenerse usando la fórmula

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

$1 \cdot - 49 - 4 (0 - 1 \cdot 3 + 3 (2 \cdot 0 - 1 \cdot 4)) + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

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

$2$ por

$0$ .

$1 \cdot - 49 - 4 (0 - 1 \cdot 3 + 3 (0 - 1 \cdot 4)) + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 4.4.2.1.2

Multiplica

$- 1$ por

$4$ .

$1 \cdot - 49 - 4 (0 - 1 \cdot 3 + 3 (0 - 4)) + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 4.4.2.2

Resta

$4$ de

$0$ .

$1 \cdot - 49 - 4 (0 - 1 \cdot 3 + 3 \cdot - 4) + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

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

$- 1$ por

$3$ .

$1 \cdot - 49 - 4 (0 - 3 + 3 \cdot - 4) + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 4.5.1.2

Multiplica

$3$ por

$- 4$ .

$1 \cdot - 49 - 4 (0 - 3 - 12) + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 4.5.2

Resta

$3$ de

$0$ .

$1 \cdot - 49 - 4 (- 3 - 12) + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 4.5.3

Resta

$12$ de

$- 3$ .

$1 \cdot - 49 - 4 \cdot - 15 + 5 | \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} | + 0$

Paso 5

Evalúa

$| \begin{array}{c} 0 & 2 & 3 \\ 2 & 5 & 1 \\ 1 & 5 & 2 \end{array} |$ .

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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.

$| \begin{array}{c} 5 & 1 \\ 5 & 2 \end{array} |$

Paso 5.1.4

Multiply element

$a_{11}$ by its cofactor.

$0 | \begin{array}{c} 5 & 1 \\ 5 & 2 \end{array} |$

Paso 5.1.5

The minor for

$a_{12}$ is the determinant with row

$1$ and column

$2$ deleted.

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

Paso 5.1.6

Multiply element

$a_{12}$ by its cofactor.

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

Paso 5.1.7

The minor for

$a_{13}$ is the determinant with row

$1$ and column

$3$ deleted.

$| \begin{array}{c} 2 & 5 \\ 1 & 5 \end{array} |$

Paso 5.1.8

Multiply element

$a_{13}$ by its cofactor.

$3 | \begin{array}{c} 2 & 5 \\ 1 & 5 \end{array} |$

Paso 5.1.9

Add the terms together.

$1 \cdot - 49 - 4 \cdot - 15 + 5 (0 | \begin{array}{c} 5 & 1 \\ 5 & 2 \end{array} | - 2 | \begin{array}{c} 2 & 1 \\ 1 & 2 \end{array} | + 3 | \begin{array}{c} 2 & 5 \\ 1 & 5 \end{array} |) + 0$

Paso 5.2

Multiplica

$0$ por

$| \begin{array}{c} 5 & 1 \\ 5 & 2 \end{array} |$ .

$1 \cdot - 49 - 4 \cdot - 15 + 5 (0 - 2 | \begin{array}{c} 2 & 1 \\ 1 & 2 \end{array} | + 3 | \begin{array}{c} 2 & 5 \\ 1 & 5 \end{array} |) + 0$

Paso 5.3

Evalúa

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

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Paso 5.3.1

El determinante de una matriz

$2 \times 2$ puede obtenerse usando la fórmula

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

$1 \cdot - 49 - 4 \cdot - 15 + 5 (0 - 2 (2 \cdot 2 - 1 \cdot 1) + 3 | \begin{array}{c} 2 & 5 \\ 1 & 5 \end{array} |) + 0$

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

$2$ por

$2$ .

$1 \cdot - 49 - 4 \cdot - 15 + 5 (0 - 2 (4 - 1 \cdot 1) + 3 | \begin{array}{c} 2 & 5 \\ 1 & 5 \end{array} |) + 0$

Paso 5.3.2.1.2

Multiplica

$- 1$ por

$1$ .

$1 \cdot - 49 - 4 \cdot - 15 + 5 (0 - 2 (4 - 1) + 3 | \begin{array}{c} 2 & 5 \\ 1 & 5 \end{array} |) + 0$

Paso 5.3.2.2

Resta

$1$ de

$4$ .

$1 \cdot - 49 - 4 \cdot - 15 + 5 (0 - 2 \cdot 3 + 3 | \begin{array}{c} 2 & 5 \\ 1 & 5 \end{array} |) + 0$

Paso 5.4

Evalúa

$| \begin{array}{c} 2 & 5 \\ 1 & 5 \end{array} |$ .

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Paso 5.4.1

El determinante de una matriz

$2 \times 2$ puede obtenerse usando la fórmula

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

$1 \cdot - 49 - 4 \cdot - 15 + 5 (0 - 2 \cdot 3 + 3 (2 \cdot 5 - 1 \cdot 5)) + 0$

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

$2$ por

$5$ .

$1 \cdot - 49 - 4 \cdot - 15 + 5 (0 - 2 \cdot 3 + 3 (10 - 1 \cdot 5)) + 0$

Paso 5.4.2.1.2

Multiplica

$- 1$ por

$5$ .

$1 \cdot - 49 - 4 \cdot - 15 + 5 (0 - 2 \cdot 3 + 3 (10 - 5)) + 0$

Paso 5.4.2.2

Resta

$5$ de

$10$ .

$1 \cdot - 49 - 4 \cdot - 15 + 5 (0 - 2 \cdot 3 + 3 \cdot 5) + 0$

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

$- 2$ por

$3$ .

$1 \cdot - 49 - 4 \cdot - 15 + 5 (0 - 6 + 3 \cdot 5) + 0$

Paso 5.5.1.2

Multiplica

$3$ por

$5$ .

$1 \cdot - 49 - 4 \cdot - 15 + 5 (0 - 6 + 15) + 0$

Paso 5.5.2

Resta

$6$ de

$0$ .

$1 \cdot - 49 - 4 \cdot - 15 + 5 (- 6 + 15) + 0$

Paso 5.5.3

Suma

$- 6$ y

$15$ .

$1 \cdot - 49 - 4 \cdot - 15 + 5 \cdot 9 + 0$

Paso 6

Simplifica el determinante.

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Paso 6.1

Simplifica cada término.

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Paso 6.1.1

Multiplica

$- 49$ por

$1$ .

$- 49 - 4 \cdot - 15 + 5 \cdot 9 + 0$

Paso 6.1.2

Multiplica

$- 4$ por

$- 15$ .

$- 49 + 60 + 5 \cdot 9 + 0$

Paso 6.1.3

Multiplica

$5$ por

$9$ .

$- 49 + 60 + 45 + 0$

Paso 6.2

Suma

$- 49$ y

$60$ .

$11 + 45 + 0$

Paso 6.3

Suma

$11$ y

$45$ .

$56 + 0$

Paso 6.4

Suma

$56$ y

$0$ .

$56$

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