Álgebra linear Exemplos

$[\begin{array}{c} \sqrt{5} & 2 \sqrt{6} & - 1 \\ - 2 \sqrt{5} & \sqrt{6} & 2 \\ \sqrt{5} & 0 & 5 \end{array}]$

Etapa 1

Find the determinant.

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Etapa 1.1

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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Etapa 1.1.1

Consider the corresponding sign chart.

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

Etapa 1.1.2

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

Etapa 1.1.3

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

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

Etapa 1.1.4

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

$- 2 \sqrt{6} | \begin{array}{c} - 2 \sqrt{5} & 2 \\ \sqrt{5} & 5 \end{array} |$

Etapa 1.1.5

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

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

Etapa 1.1.6

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

$\sqrt{6} | \begin{array}{c} \sqrt{5} & - 1 \\ \sqrt{5} & 5 \end{array} |$

Etapa 1.1.7

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

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

Etapa 1.1.8

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

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

Etapa 1.1.9

Add the terms together.

$- 2 \sqrt{6} | \begin{array}{c} - 2 \sqrt{5} & 2 \\ \sqrt{5} & 5 \end{array} | + \sqrt{6} | \begin{array}{c} \sqrt{5} & - 1 \\ \sqrt{5} & 5 \end{array} | + 0 | \begin{array}{c} \sqrt{5} & - 1 \\ - 2 \sqrt{5} & 2 \end{array} |$

Etapa 1.2

Multiplique $0$ por $| \begin{array}{c} \sqrt{5} & - 1 \\ - 2 \sqrt{5} & 2 \end{array} |$ .

$- 2 \sqrt{6} | \begin{array}{c} - 2 \sqrt{5} & 2 \\ \sqrt{5} & 5 \end{array} | + \sqrt{6} | \begin{array}{c} \sqrt{5} & - 1 \\ \sqrt{5} & 5 \end{array} | + 0$

Etapa 1.3

Avalie $| \begin{array}{c} - 2 \sqrt{5} & 2 \\ \sqrt{5} & 5 \end{array} |$ .

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Etapa 1.3.1

O determinante de uma matriz $2 \times 2$ pode ser encontrado ao usar a fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 2 \sqrt{6} (- 2 \sqrt{5} \cdot 5 - \sqrt{5} \cdot 2) + \sqrt{6} | \begin{array}{c} \sqrt{5} & - 1 \\ \sqrt{5} & 5 \end{array} | + 0$

Etapa 1.3.2

Simplifique o determinante.

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Etapa 1.3.2.1

Simplifique cada termo.

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Etapa 1.3.2.1.1

Multiplique $5$ por $- 2$ .

$- 2 \sqrt{6} (- 10 \sqrt{5} - \sqrt{5} \cdot 2) + \sqrt{6} | \begin{array}{c} \sqrt{5} & - 1 \\ \sqrt{5} & 5 \end{array} | + 0$

Etapa 1.3.2.1.2

Multiplique $2$ por $- 1$ .

$- 2 \sqrt{6} (- 10 \sqrt{5} - 2 \sqrt{5}) + \sqrt{6} | \begin{array}{c} \sqrt{5} & - 1 \\ \sqrt{5} & 5 \end{array} | + 0$

Etapa 1.3.2.2

Subtraia $2 \sqrt{5}$ de $- 10 \sqrt{5}$ .

$- 2 \sqrt{6} (- 12 \sqrt{5}) + \sqrt{6} | \begin{array}{c} \sqrt{5} & - 1 \\ \sqrt{5} & 5 \end{array} | + 0$

Etapa 1.4

Avalie $| \begin{array}{c} \sqrt{5} & - 1 \\ \sqrt{5} & 5 \end{array} |$ .

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Etapa 1.4.1

O determinante de uma matriz $2 \times 2$ pode ser encontrado ao usar a fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 2 \sqrt{6} (- 12 \sqrt{5}) + \sqrt{6} (\sqrt{5} \cdot 5 - \sqrt{5} \cdot - 1) + 0$

Etapa 1.4.2

Simplifique o determinante.

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Etapa 1.4.2.1

Simplifique cada termo.

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Etapa 1.4.2.1.1

Mova $5$ para a esquerda de $\sqrt{5}$ .

$- 2 \sqrt{6} (- 12 \sqrt{5}) + \sqrt{6} (5 \cdot \sqrt{5} - \sqrt{5} \cdot - 1) + 0$

Etapa 1.4.2.1.2

Multiplique $- \sqrt{5} \cdot - 1$ .

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Etapa 1.4.2.1.2.1

Multiplique $- 1$ por $- 1$ .

$- 2 \sqrt{6} (- 12 \sqrt{5}) + \sqrt{6} (5 \sqrt{5} + 1 \sqrt{5}) + 0$

Etapa 1.4.2.1.2.2

Multiplique $\sqrt{5}$ por $1$ .

$- 2 \sqrt{6} (- 12 \sqrt{5}) + \sqrt{6} (5 \sqrt{5} + \sqrt{5}) + 0$

Etapa 1.4.2.2

Some $5 \sqrt{5}$ e $\sqrt{5}$ .

$- 2 \sqrt{6} (- 12 \sqrt{5}) + \sqrt{6} (6 \sqrt{5}) + 0$

Etapa 1.5

Simplifique o determinante.

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Etapa 1.5.1

Simplifique cada termo.

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Etapa 1.5.1.1

Multiplique $- 2 \sqrt{6} (- 12 \sqrt{5})$ .

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Etapa 1.5.1.1.1

Multiplique $- 12$ por $- 2$ .

$24 \sqrt{6} \sqrt{5} + \sqrt{6} (6 \sqrt{5}) + 0$

Etapa 1.5.1.1.2

Combine usando a regra do produto para radicais.

$24 \sqrt{5 \cdot 6} + \sqrt{6} (6 \sqrt{5}) + 0$

Etapa 1.5.1.1.3

Multiplique $5$ por $6$ .

$24 \sqrt{30} + \sqrt{6} (6 \sqrt{5}) + 0$

Etapa 1.5.1.2

Multiplique $\sqrt{6} (6 \sqrt{5})$ .

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Etapa 1.5.1.2.1

Combine usando a regra do produto para radicais.

$24 \sqrt{30} + 6 \sqrt{6 \cdot 5} + 0$

Etapa 1.5.1.2.2

Multiplique $6$ por $5$ .

$24 \sqrt{30} + 6 \sqrt{30} + 0$

Etapa 1.5.2

Some $24 \sqrt{30}$ e $6 \sqrt{30}$ .

$30 \sqrt{30} + 0$

Etapa 1.5.3

Some $30 \sqrt{30}$ e $0$ .

$30 \sqrt{30}$

Etapa 2

Since the determinant is non-zero, the inverse exists.

Etapa 3

Set up a $3 \times 6$ matrix where the left half is the original matrix and the right half is its identity matrix.

$[\begin{array}{c} \sqrt{5} & 2 \sqrt{6} & - 1 & 1 & 0 & 0 \\ - 2 \sqrt{5} & \sqrt{6} & 2 & 0 & 1 & 0 \\ \sqrt{5} & 0 & 5 & 0 & 0 & 1 \end{array}]$

Etapa 4

Encontre a forma escalonada reduzida por linhas.

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Etapa 4.1

Multiply each element of $R_{1}$ by $\frac{1}{\sqrt{5}}$ to make the entry at $1, 1$ a $1$ .

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Etapa 4.1.1

Multiply each element of $R_{1}$ by $\frac{1}{\sqrt{5}}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{\sqrt{5}}{\sqrt{5}} & \frac{2 \sqrt{6}}{\sqrt{5}} & \frac{- 1}{\sqrt{5}} & \frac{1}{\sqrt{5}} & \frac{0}{\sqrt{5}} & \frac{0}{\sqrt{5}} \\ - 2 \sqrt{5} & \sqrt{6} & 2 & 0 & 1 & 0 \\ \sqrt{5} & 0 & 5 & 0 & 0 & 1 \end{array}]$

Etapa 4.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2 \sqrt{30}}{5} & - \frac{\sqrt{5}}{5} & \frac{\sqrt{5}}{5} & 0 & 0 \\ - 2 \sqrt{5} & \sqrt{6} & 2 & 0 & 1 & 0 \\ \sqrt{5} & 0 & 5 & 0 & 0 & 1 \end{array}]$

Etapa 4.2

Perform the row operation $R_{2} = R_{2} + 2 \sqrt{5} R_{1}$ to make the entry at $2, 1$ a $0$ .

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Etapa 4.2.1

Perform the row operation $R_{2} = R_{2} + 2 \sqrt{5} R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{2 \sqrt{30}}{5} & - \frac{\sqrt{5}}{5} & \frac{\sqrt{5}}{5} & 0 & 0 \\ - 2 \sqrt{5} + 2 \sqrt{5} \cdot 1 & \sqrt{6} + 2 \sqrt{5} \frac{2 \sqrt{30}}{5} & 2 + 2 \sqrt{5} (- \frac{\sqrt{5}}{5}) & 0 + 2 \sqrt{5} \frac{\sqrt{5}}{5} & 1 + 2 \sqrt{5} \cdot 0 & 0 + 2 \sqrt{5} \cdot 0 \\ \sqrt{5} & 0 & 5 & 0 & 0 & 1 \end{array}]$

Etapa 4.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2 \sqrt{30}}{5} & - \frac{\sqrt{5}}{5} & \frac{\sqrt{5}}{5} & 0 & 0 \\ 0 & 5 \sqrt{6} & 0 & 2 & 1 & 0 \\ \sqrt{5} & 0 & 5 & 0 & 0 & 1 \end{array}]$

Etapa 4.3

Perform the row operation $R_{3} = R_{3} - \sqrt{5} R_{1}$ to make the entry at $3, 1$ a $0$ .

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Etapa 4.3.1

Perform the row operation $R_{3} = R_{3} - \sqrt{5} R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{2 \sqrt{30}}{5} & - \frac{\sqrt{5}}{5} & \frac{\sqrt{5}}{5} & 0 & 0 \\ 0 & 5 \sqrt{6} & 0 & 2 & 1 & 0 \\ \sqrt{5} - \sqrt{5} \cdot 1 & 0 - \sqrt{5} \frac{2 \sqrt{30}}{5} & 5 - \sqrt{5} (- \frac{\sqrt{5}}{5}) & 0 - \sqrt{5} \frac{\sqrt{5}}{5} & 0 - \sqrt{5} \cdot 0 & 1 - \sqrt{5} \cdot 0 \end{array}]$

Etapa 4.3.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2 \sqrt{30}}{5} & - \frac{\sqrt{5}}{5} & \frac{\sqrt{5}}{5} & 0 & 0 \\ 0 & 5 \sqrt{6} & 0 & 2 & 1 & 0 \\ 0 & - 2 \sqrt{6} & 6 & - 1 & 0 & 1 \end{array}]$

Etapa 4.4

Multiply each element of $R_{2}$ by $\frac{1}{5 \sqrt{6}}$ to make the entry at $2, 2$ a $1$ .

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Etapa 4.4.1

Multiply each element of $R_{2}$ by $\frac{1}{5 \sqrt{6}}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{2 \sqrt{30}}{5} & - \frac{\sqrt{5}}{5} & \frac{\sqrt{5}}{5} & 0 & 0 \\ \frac{0}{5 \sqrt{6}} & \frac{5 \sqrt{6}}{5 \sqrt{6}} & \frac{0}{5 \sqrt{6}} & \frac{2}{5 \sqrt{6}} & \frac{1}{5 \sqrt{6}} & \frac{0}{5 \sqrt{6}} \\ 0 & - 2 \sqrt{6} & 6 & - 1 & 0 & 1 \end{array}]$

Etapa 4.4.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2 \sqrt{30}}{5} & - \frac{\sqrt{5}}{5} & \frac{\sqrt{5}}{5} & 0 & 0 \\ 0 & 1 & 0 & \frac{\sqrt{6}}{15} & \frac{\sqrt{6}}{30} & 0 \\ 0 & - 2 \sqrt{6} & 6 & - 1 & 0 & 1 \end{array}]$

Etapa 4.5

Perform the row operation $R_{3} = R_{3} + 2 \sqrt{6} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 4.5.1

Perform the row operation $R_{3} = R_{3} + 2 \sqrt{6} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{2 \sqrt{30}}{5} & - \frac{\sqrt{5}}{5} & \frac{\sqrt{5}}{5} & 0 & 0 \\ 0 & 1 & 0 & \frac{\sqrt{6}}{15} & \frac{\sqrt{6}}{30} & 0 \\ 0 + 2 \sqrt{6} \cdot 0 & - 2 \sqrt{6} + 2 \sqrt{6} \cdot 1 & 6 + 2 \sqrt{6} \cdot 0 & - 1 + 2 \sqrt{6} \frac{\sqrt{6}}{15} & 0 + 2 \sqrt{6} \frac{\sqrt{6}}{30} & 1 + 2 \sqrt{6} \cdot 0 \end{array}]$

Etapa 4.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2 \sqrt{30}}{5} & - \frac{\sqrt{5}}{5} & \frac{\sqrt{5}}{5} & 0 & 0 \\ 0 & 1 & 0 & \frac{\sqrt{6}}{15} & \frac{\sqrt{6}}{30} & 0 \\ 0 & 0 & 6 & - \frac{1}{5} & \frac{2}{5} & 1 \end{array}]$

Etapa 4.6

Multiply each element of $R_{3}$ by $\frac{1}{6}$ to make the entry at $3, 3$ a $1$ .

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Etapa 4.6.1

Multiply each element of $R_{3}$ by $\frac{1}{6}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{2 \sqrt{30}}{5} & - \frac{\sqrt{5}}{5} & \frac{\sqrt{5}}{5} & 0 & 0 \\ 0 & 1 & 0 & \frac{\sqrt{6}}{15} & \frac{\sqrt{6}}{30} & 0 \\ \frac{0}{6} & \frac{0}{6} & \frac{6}{6} & \frac{- \frac{1}{5}}{6} & \frac{\frac{2}{5}}{6} & \frac{1}{6} \end{array}]$

Etapa 4.6.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2 \sqrt{30}}{5} & - \frac{\sqrt{5}}{5} & \frac{\sqrt{5}}{5} & 0 & 0 \\ 0 & 1 & 0 & \frac{\sqrt{6}}{15} & \frac{\sqrt{6}}{30} & 0 \\ 0 & 0 & 1 & - \frac{1}{30} & \frac{1}{15} & \frac{1}{6} \end{array}]$

Etapa 4.7

Perform the row operation $R_{1} = R_{1} + \frac{\sqrt{5}}{5} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Etapa 4.7.1

Perform the row operation $R_{1} = R_{1} + \frac{\sqrt{5}}{5} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 + \frac{\sqrt{5}}{5} \cdot 0 & \frac{2 \sqrt{30}}{5} + \frac{\sqrt{5}}{5} \cdot 0 & - \frac{\sqrt{5}}{5} + \frac{\sqrt{5}}{5} \cdot 1 & \frac{\sqrt{5}}{5} + \frac{\sqrt{5}}{5} (- \frac{1}{30}) & 0 + \frac{\sqrt{5}}{5} \cdot \frac{1}{15} & 0 + \frac{\sqrt{5}}{5} \cdot \frac{1}{6} \\ 0 & 1 & 0 & \frac{\sqrt{6}}{15} & \frac{\sqrt{6}}{30} & 0 \\ 0 & 0 & 1 & - \frac{1}{30} & \frac{1}{15} & \frac{1}{6} \end{array}]$

Etapa 4.7.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2 \sqrt{30}}{5} & 0 & \frac{29 \sqrt{5}}{150} & \frac{\sqrt{5}}{75} & \frac{\sqrt{5}}{30} \\ 0 & 1 & 0 & \frac{\sqrt{6}}{15} & \frac{\sqrt{6}}{30} & 0 \\ 0 & 0 & 1 & - \frac{1}{30} & \frac{1}{15} & \frac{1}{6} \end{array}]$

Etapa 4.8

Perform the row operation $R_{1} = R_{1} - \frac{2 \sqrt{30}}{5} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Etapa 4.8.1

Perform the row operation $R_{1} = R_{1} - \frac{2 \sqrt{30}}{5} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{2 \sqrt{30}}{5} \cdot 0 & \frac{2 \sqrt{30}}{5} - \frac{2 \sqrt{30}}{5} \cdot 1 & 0 - \frac{2 \sqrt{30}}{5} \cdot 0 & \frac{29 \sqrt{5}}{150} - \frac{2 \sqrt{30}}{5} \cdot \frac{\sqrt{6}}{15} & \frac{\sqrt{5}}{75} - \frac{2 \sqrt{30}}{5} \cdot \frac{\sqrt{6}}{30} & \frac{\sqrt{5}}{30} - \frac{2 \sqrt{30}}{5} \cdot 0 \\ 0 & 1 & 0 & \frac{\sqrt{6}}{15} & \frac{\sqrt{6}}{30} & 0 \\ 0 & 0 & 1 & - \frac{1}{30} & \frac{1}{15} & \frac{1}{6} \end{array}]$

Etapa 4.8.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & \frac{\sqrt{5}}{30} & - \frac{\sqrt{5}}{15} & \frac{\sqrt{5}}{30} \\ 0 & 1 & 0 & \frac{\sqrt{6}}{15} & \frac{\sqrt{6}}{30} & 0 \\ 0 & 0 & 1 & - \frac{1}{30} & \frac{1}{15} & \frac{1}{6} \end{array}]$

Etapa 5

The right half of the reduced row echelon form is the inverse.

$[\begin{array}{c} \frac{\sqrt{5}}{30} & - \frac{\sqrt{5}}{15} & \frac{\sqrt{5}}{30} \\ \frac{\sqrt{6}}{15} & \frac{\sqrt{6}}{30} & 0 \\ - \frac{1}{30} & \frac{1}{15} & \frac{1}{6} \end{array}]$