Álgebra linear Exemplos

$[\begin{array}{c} 13 & 8 & 15 \\ 6 & 0 & 7 \\ 7 & 9 & 8 \end{array}]$

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

$| \begin{array}{c} 8 & 15 \\ 9 & 8 \end{array} |$

Etapa 1.1.4

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

$- 6 | \begin{array}{c} 8 & 15 \\ 9 & 8 \end{array} |$

Etapa 1.1.5

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

$| \begin{array}{c} 13 & 15 \\ 7 & 8 \end{array} |$

Etapa 1.1.6

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

$0 | \begin{array}{c} 13 & 15 \\ 7 & 8 \end{array} |$

Etapa 1.1.7

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

$| \begin{array}{c} 13 & 8 \\ 7 & 9 \end{array} |$

Etapa 1.1.8

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

$- 7 | \begin{array}{c} 13 & 8 \\ 7 & 9 \end{array} |$

Etapa 1.1.9

Add the terms together.

$- 6 | \begin{array}{c} 8 & 15 \\ 9 & 8 \end{array} | + 0 | \begin{array}{c} 13 & 15 \\ 7 & 8 \end{array} | - 7 | \begin{array}{c} 13 & 8 \\ 7 & 9 \end{array} |$

Etapa 1.2

Multiplique $0$ por $| \begin{array}{c} 13 & 15 \\ 7 & 8 \end{array} |$ .

$- 6 | \begin{array}{c} 8 & 15 \\ 9 & 8 \end{array} | + 0 - 7 | \begin{array}{c} 13 & 8 \\ 7 & 9 \end{array} |$

Etapa 1.3

Avalie $| \begin{array}{c} 8 & 15 \\ 9 & 8 \end{array} |$ .

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Etapa 1.3.1

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

$- 6 (8 \cdot 8 - 9 \cdot 15) + 0 - 7 | \begin{array}{c} 13 & 8 \\ 7 & 9 \end{array} |$

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 $8$ por $8$ .

$- 6 (64 - 9 \cdot 15) + 0 - 7 | \begin{array}{c} 13 & 8 \\ 7 & 9 \end{array} |$

Etapa 1.3.2.1.2

Multiplique $- 9$ por $15$ .

$- 6 (64 - 135) + 0 - 7 | \begin{array}{c} 13 & 8 \\ 7 & 9 \end{array} |$

Etapa 1.3.2.2

Subtraia $135$ de $64$ .

$- 6 \cdot - 71 + 0 - 7 | \begin{array}{c} 13 & 8 \\ 7 & 9 \end{array} |$

Etapa 1.4

Avalie $| \begin{array}{c} 13 & 8 \\ 7 & 9 \end{array} |$ .

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Etapa 1.4.1

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

$- 6 \cdot - 71 + 0 - 7 (13 \cdot 9 - 7 \cdot 8)$

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

Multiplique $13$ por $9$ .

$- 6 \cdot - 71 + 0 - 7 (117 - 7 \cdot 8)$

Etapa 1.4.2.1.2

Multiplique $- 7$ por $8$ .

$- 6 \cdot - 71 + 0 - 7 (117 - 56)$

Etapa 1.4.2.2

Subtraia $56$ de $117$ .

$- 6 \cdot - 71 + 0 - 7 \cdot 61$

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 $- 6$ por $- 71$ .

$426 + 0 - 7 \cdot 61$

Etapa 1.5.1.2

Multiplique $- 7$ por $61$ .

$426 + 0 - 427$

Etapa 1.5.2

Some $426$ e $0$ .

$426 - 427$

Etapa 1.5.3

Subtraia $427$ de $426$ .

$- 1$

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.

$[\begin{array}{c} 13 & 8 & 15 & 1 & 0 & 0 \\ 6 & 0 & 7 & 0 & 1 & 0 \\ 7 & 9 & 8 & 0 & 0 & 1 \end{array}]$

Etapa 4

Encontre a forma escalonada reduzida por linhas.

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Etapa 4.1

Multiply each element of $R_{1}$ by $\frac{1}{13}$ to make the entry at $1, 1$ a $1$ .

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Etapa 4.1.1

Multiply each element of $R_{1}$ by $\frac{1}{13}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{13}{13} & \frac{8}{13} & \frac{15}{13} & \frac{1}{13} & \frac{0}{13} & \frac{0}{13} \\ 6 & 0 & 7 & 0 & 1 & 0 \\ 7 & 9 & 8 & 0 & 0 & 1 \end{array}]$

Etapa 4.1.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & \frac{8}{13} & \frac{15}{13} & \frac{1}{13} & 0 & 0 \\ 6 & 0 & 7 & 0 & 1 & 0 \\ 7 & 9 & 8 & 0 & 0 & 1 \end{array}]$

Etapa 4.2

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Etapa 4.2.1

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{8}{13} & \frac{15}{13} & \frac{1}{13} & 0 & 0 \\ 6 - 6 \cdot 1 & 0 - 6 (\frac{8}{13}) & 7 - 6 (\frac{15}{13}) & 0 - 6 (\frac{1}{13}) & 1 - 6 \cdot 0 & 0 - 6 \cdot 0 \\ 7 & 9 & 8 & 0 & 0 & 1 \end{array}]$

Etapa 4.2.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{8}{13} & \frac{15}{13} & \frac{1}{13} & 0 & 0 \\ 0 & - \frac{48}{13} & \frac{1}{13} & - \frac{6}{13} & 1 & 0 \\ 7 & 9 & 8 & 0 & 0 & 1 \end{array}]$

Etapa 4.3

Perform the row operation $R_{3} = R_{3} - 7 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Etapa 4.3.1

Perform the row operation $R_{3} = R_{3} - 7 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{8}{13} & \frac{15}{13} & \frac{1}{13} & 0 & 0 \\ 0 & - \frac{48}{13} & \frac{1}{13} & - \frac{6}{13} & 1 & 0 \\ 7 - 7 \cdot 1 & 9 - 7 (\frac{8}{13}) & 8 - 7 (\frac{15}{13}) & 0 - 7 (\frac{1}{13}) & 0 - 7 \cdot 0 & 1 - 7 \cdot 0 \end{array}]$

Etapa 4.3.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{8}{13} & \frac{15}{13} & \frac{1}{13} & 0 & 0 \\ 0 & - \frac{48}{13} & \frac{1}{13} & - \frac{6}{13} & 1 & 0 \\ 0 & \frac{61}{13} & - \frac{1}{13} & - \frac{7}{13} & 0 & 1 \end{array}]$

Etapa 4.4

Multiply each element of $R_{2}$ by $- \frac{13}{48}$ to make the entry at $2, 2$ a $1$ .

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Etapa 4.4.1

Multiply each element of $R_{2}$ by $- \frac{13}{48}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{8}{13} & \frac{15}{13} & \frac{1}{13} & 0 & 0 \\ - \frac{13}{48} \cdot 0 & - \frac{13}{48} (- \frac{48}{13}) & - \frac{13}{48} \cdot \frac{1}{13} & - \frac{13}{48} (- \frac{6}{13}) & - \frac{13}{48} \cdot 1 & - \frac{13}{48} \cdot 0 \\ 0 & \frac{61}{13} & - \frac{1}{13} & - \frac{7}{13} & 0 & 1 \end{array}]$

Etapa 4.4.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{8}{13} & \frac{15}{13} & \frac{1}{13} & 0 & 0 \\ 0 & 1 & - \frac{1}{48} & \frac{1}{8} & - \frac{13}{48} & 0 \\ 0 & \frac{61}{13} & - \frac{1}{13} & - \frac{7}{13} & 0 & 1 \end{array}]$

Etapa 4.5

Perform the row operation $R_{3} = R_{3} - \frac{61}{13} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Etapa 4.5.1

Perform the row operation $R_{3} = R_{3} - \frac{61}{13} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{8}{13} & \frac{15}{13} & \frac{1}{13} & 0 & 0 \\ 0 & 1 & - \frac{1}{48} & \frac{1}{8} & - \frac{13}{48} & 0 \\ 0 - \frac{61}{13} \cdot 0 & \frac{61}{13} - \frac{61}{13} \cdot 1 & - \frac{1}{13} - \frac{61}{13} (- \frac{1}{48}) & - \frac{7}{13} - \frac{61}{13} \cdot \frac{1}{8} & 0 - \frac{61}{13} (- \frac{13}{48}) & 1 - \frac{61}{13} \cdot 0 \end{array}]$

Etapa 4.5.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{8}{13} & \frac{15}{13} & \frac{1}{13} & 0 & 0 \\ 0 & 1 & - \frac{1}{48} & \frac{1}{8} & - \frac{13}{48} & 0 \\ 0 & 0 & \frac{1}{48} & - \frac{9}{8} & \frac{61}{48} & 1 \end{array}]$

Etapa 4.6

Multiply each element of $R_{3}$ by $48$ to make the entry at $3, 3$ a $1$ .

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Etapa 4.6.1

Multiply each element of $R_{3}$ by $48$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{8}{13} & \frac{15}{13} & \frac{1}{13} & 0 & 0 \\ 0 & 1 & - \frac{1}{48} & \frac{1}{8} & - \frac{13}{48} & 0 \\ 48 \cdot 0 & 48 \cdot 0 & 48 (\frac{1}{48}) & 48 (- \frac{9}{8}) & 48 (\frac{61}{48}) & 48 \cdot 1 \end{array}]$

Etapa 4.6.2

Simplifique $R_{3}$ .

$[\begin{array}{c} 1 & \frac{8}{13} & \frac{15}{13} & \frac{1}{13} & 0 & 0 \\ 0 & 1 & - \frac{1}{48} & \frac{1}{8} & - \frac{13}{48} & 0 \\ 0 & 0 & 1 & - 54 & 61 & 48 \end{array}]$

Etapa 4.7

Perform the row operation $R_{2} = R_{2} + \frac{1}{48} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Etapa 4.7.1

Perform the row operation $R_{2} = R_{2} + \frac{1}{48} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{8}{13} & \frac{15}{13} & \frac{1}{13} & 0 & 0 \\ 0 + \frac{1}{48} \cdot 0 & 1 + \frac{1}{48} \cdot 0 & - \frac{1}{48} + \frac{1}{48} \cdot 1 & \frac{1}{8} + \frac{1}{48} \cdot - 54 & - \frac{13}{48} + \frac{1}{48} \cdot 61 & 0 + \frac{1}{48} \cdot 48 \\ 0 & 0 & 1 & - 54 & 61 & 48 \end{array}]$

Etapa 4.7.2

Simplifique $R_{2}$ .

$[\begin{array}{c} 1 & \frac{8}{13} & \frac{15}{13} & \frac{1}{13} & 0 & 0 \\ 0 & 1 & 0 & - 1 & 1 & 1 \\ 0 & 0 & 1 & - 54 & 61 & 48 \end{array}]$

Etapa 4.8

Perform the row operation $R_{1} = R_{1} - \frac{15}{13} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Etapa 4.8.1

Perform the row operation $R_{1} = R_{1} - \frac{15}{13} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{15}{13} \cdot 0 & \frac{8}{13} - \frac{15}{13} \cdot 0 & \frac{15}{13} - \frac{15}{13} \cdot 1 & \frac{1}{13} - \frac{15}{13} \cdot - 54 & 0 - \frac{15}{13} \cdot 61 & 0 - \frac{15}{13} \cdot 48 \\ 0 & 1 & 0 & - 1 & 1 & 1 \\ 0 & 0 & 1 & - 54 & 61 & 48 \end{array}]$

Etapa 4.8.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & \frac{8}{13} & 0 & \frac{811}{13} & - \frac{915}{13} & - \frac{720}{13} \\ 0 & 1 & 0 & - 1 & 1 & 1 \\ 0 & 0 & 1 & - 54 & 61 & 48 \end{array}]$

Etapa 4.9

Perform the row operation $R_{1} = R_{1} - \frac{8}{13} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Etapa 4.9.1

Perform the row operation $R_{1} = R_{1} - \frac{8}{13} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{8}{13} \cdot 0 & \frac{8}{13} - \frac{8}{13} \cdot 1 & 0 - \frac{8}{13} \cdot 0 & \frac{811}{13} - \frac{8}{13} \cdot - 1 & - \frac{915}{13} - \frac{8}{13} \cdot 1 & - \frac{720}{13} - \frac{8}{13} \cdot 1 \\ 0 & 1 & 0 & - 1 & 1 & 1 \\ 0 & 0 & 1 & - 54 & 61 & 48 \end{array}]$

Etapa 4.9.2

Simplifique $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 63 & - 71 & - 56 \\ 0 & 1 & 0 & - 1 & 1 & 1 \\ 0 & 0 & 1 & - 54 & 61 & 48 \end{array}]$

Etapa 5

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

$[\begin{array}{c} 63 & - 71 & - 56 \\ - 1 & 1 & 1 \\ - 54 & 61 & 48 \end{array}]$