Matemática discreta Exemplos

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

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

Etapa 1

Find the determinant.

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Etapa 1.1

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.

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Etapa 1.1.1

Consider the corresponding sign chart.

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

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

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}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

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

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

Etapa 1.1.4

Multiply element

a_{12}

$a_{12}$ by its cofactor.

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

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

Etapa 1.1.5

The minor for

a_{22}

$a_{22}$ is the determinant with row

2

$2$ and column

2

$2$ deleted.

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

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

Etapa 1.1.6

Multiply element

a_{22}

$a_{22}$ by its cofactor.

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

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

Etapa 1.1.7

The minor for

a_{32}

$a_{32}$ is the determinant with row

3

$3$ and column

2

$2$ deleted.

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

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

Etapa 1.1.8

Multiply element

a_{32}

$a_{32}$ by its cofactor.

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

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

Etapa 1.1.9

Add the terms together.

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

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

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

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

Etapa 1.2

Multiplique

0

$0$ por

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

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

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

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

Etapa 1.3

Multiplique

0

$0$ por

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

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

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

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

Etapa 1.4

Avalie

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

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

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Etapa 1.4.1

O determinante de uma matriz

2 \times 2

$2 \times 2$ pode ser encontrado ao usar a fórmula

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

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

0 - 2 (1 \cdot 0 - 3 \cdot 1) + 0

$0 - 2 (1 \cdot 0 - 3 \cdot 1) + 0$

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

Multiplique

0

$0$ por

1

$1$ .

0 - 2 (0 - 3 \cdot 1) + 0

$0 - 2 (0 - 3 \cdot 1) + 0$

Etapa 1.4.2.1.2

Multiplique

- 3

$- 3$ por

1

$1$ .

0 - 2 (0 - 3) + 0

$0 - 2 (0 - 3) + 0$

0 - 2 (0 - 3) + 0

$0 - 2 (0 - 3) + 0$

Etapa 1.4.2.2

Subtraia

3

$3$ de

0

$0$ .

0 - 2 \cdot - 3 + 0

$0 - 2 \cdot - 3 + 0$

0 - 2 \cdot - 3 + 0

$0 - 2 \cdot - 3 + 0$

0 - 2 \cdot - 3 + 0

$0 - 2 \cdot - 3 + 0$

Etapa 1.5

Simplifique o determinante.

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Etapa 1.5.1

Multiplique

- 2

$- 2$ por

- 3

$- 3$ .

0 + 6 + 0

$0 + 6 + 0$

Etapa 1.5.2

Some

0

$0$ e

6

$6$ .

6 + 0

$6 + 0$

Etapa 1.5.3

Some

6

$6$ e

0

$0$ .

6

$6$

6

$6$

6

$6$

Etapa 2

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

Etapa 3

Set up a

3 \times 6

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

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

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

Etapa 4

Encontre a forma escalonada reduzida por linhas.

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Etapa 4.1

Perform the row operation

R_{2} = R_{2} - 2 R_{1}

$R_{2} = R_{2} - 2 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Etapa 4.1.1

Perform the row operation

R_{2} = R_{2} - 2 R_{1}

$R_{2} = R_{2} - 2 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 2 - 2 \cdot 1 & - 2 - 2 \cdot 0 & - 1 - 2 \cdot 1 & 0 - 2 \cdot 1 & 1 - 2 \cdot 0 & 0 - 2 \cdot 0 \\ 3 & 0 & 0 & 0 & 0 & 1 \end{array}]$

Etapa 4.1.2

Simplifique

R_{2}

$R_{2}$ .

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

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

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

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

Etapa 4.2

Perform the row operation

R_{3} = R_{3} - 3 R_{1}

$R_{3} = R_{3} - 3 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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Etapa 4.2.1

Perform the row operation

R_{3} = R_{3} - 3 R_{1}

$R_{3} = R_{3} - 3 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & - 2 & - 3 & - 2 & 1 & 0 \\ 3 - 3 \cdot 1 & 0 - 3 \cdot 0 & 0 - 3 \cdot 1 & 0 - 3 \cdot 1 & 0 - 3 \cdot 0 & 1 - 3 \cdot 0 \end{array}]$

Etapa 4.2.2

Simplifique

R_{3}

$R_{3}$ .

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

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

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

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

Etapa 4.3

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{1}{2}

$- \frac{1}{2}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Etapa 4.3.1

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{1}{2}

$- \frac{1}{2}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & 1 & 1 & 0 & 0 - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot - 3 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot 1 & - \frac{1}{2} \cdot 0 0 & 0 & - 3 & - 3 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot - 3 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot 1 & - \frac{1}{2} \cdot 0 \\ 0 & 0 & - 3 & - 3 & 0 & 1 \end{array}]$

Etapa 4.3.2

Simplifique

R_{2}

$R_{2}$ .

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

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & \frac{3}{2} & 1 & - \frac{1}{2} & 0 \\ 0 & 0 & - 3 & - 3 & 0 & 1 \end{array}]$

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

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & \frac{3}{2} & 1 & - \frac{1}{2} & 0 \\ 0 & 0 & - 3 & - 3 & 0 & 1 \end{array}]$

Etapa 4.4

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{1}{3}

$- \frac{1}{3}$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

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Etapa 4.4.1

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{1}{3}

$- \frac{1}{3}$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & 1 & 1 & 0 & 0 0 & 1 & \frac{3}{2} & 1 & - \frac{1}{2} & 0 - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & \frac{3}{2} & 1 & - \frac{1}{2} & 0 \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot 1 \end{array}]$

Etapa 4.4.2

Simplifique

R_{3}

$R_{3}$ .

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

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & \frac{3}{2} & 1 & - \frac{1}{2} & 0 \\ 0 & 0 & 1 & 1 & 0 & - \frac{1}{3} \end{array}]$

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

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & \frac{3}{2} & 1 & - \frac{1}{2} & 0 \\ 0 & 0 & 1 & 1 & 0 & - \frac{1}{3} \end{array}]$

Etapa 4.5

Perform the row operation

R_{2} = R_{2} - \frac{3}{2} R_{3}

$R_{2} = R_{2} - \frac{3}{2} R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

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Etapa 4.5.1

Perform the row operation

R_{2} = R_{2} - \frac{3}{2} R_{3}

$R_{2} = R_{2} - \frac{3}{2} R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & 1 & 1 & 0 & 0 0 - \frac{3}{2} \cdot 0 & 1 - \frac{3}{2} \cdot 0 & \frac{3}{2} - \frac{3}{2} \cdot 1 & 1 - \frac{3}{2} \cdot 1 & - \frac{1}{2} - \frac{3}{2} \cdot 0 & 0 - \frac{3}{2} (- \frac{1}{3}) 0 & 0 & 1 & 1 & 0 & - \frac{1}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 - \frac{3}{2} \cdot 0 & 1 - \frac{3}{2} \cdot 0 & \frac{3}{2} - \frac{3}{2} \cdot 1 & 1 - \frac{3}{2} \cdot 1 & - \frac{1}{2} - \frac{3}{2} \cdot 0 & 0 - \frac{3}{2} (- \frac{1}{3}) \\ 0 & 0 & 1 & 1 & 0 & - \frac{1}{3} \end{array}]$

Etapa 4.5.2

Simplifique

R_{2}

$R_{2}$ .

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

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & - \frac{1}{2} & - \frac{1}{2} & \frac{1}{2} \\ 0 & 0 & 1 & 1 & 0 & - \frac{1}{3} \end{array}]$

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

$[\begin{array}{c} 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & - \frac{1}{2} & - \frac{1}{2} & \frac{1}{2} \\ 0 & 0 & 1 & 1 & 0 & - \frac{1}{3} \end{array}]$

Etapa 4.6

Perform the row operation

R_{1} = R_{1} - R_{3}

$R_{1} = R_{1} - R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

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Etapa 4.6.1

Perform the row operation

R_{1} = R_{1} - R_{3}

$R_{1} = R_{1} - R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 - 0 & 0 - 0 & 1 - 1 & 1 - 1 & 0 - 0 & 0 + \frac{1}{3} 0 & 1 & 0 & - \frac{1}{2} & - \frac{1}{2} & \frac{1}{2} 0 & 0 & 1 & 1 & 0 & - \frac{1}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 - 0 & 0 - 0 & 1 - 1 & 1 - 1 & 0 - 0 & 0 + \frac{1}{3} \\ 0 & 1 & 0 & - \frac{1}{2} & - \frac{1}{2} & \frac{1}{2} \\ 0 & 0 & 1 & 1 & 0 & - \frac{1}{3} \end{array}]$

Etapa 4.6.2

Simplifique

R_{1}

$R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 & \frac{1}{3} \\ 0 & 1 & 0 & - \frac{1}{2} & - \frac{1}{2} & \frac{1}{2} \\ 0 & 0 & 1 & 1 & 0 & - \frac{1}{3} \end{array}]$

Etapa 5

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

$[\begin{array}{c} 0 & 0 & \frac{1}{3} \\ - \frac{1}{2} & - \frac{1}{2} & \frac{1}{2} \\ 1 & 0 & - \frac{1}{3} \end{array}]$