Linear Algebra Examples

$[\begin{array}{c} - \frac{4}{7} & \frac{2}{7} & \frac{1}{7} & \frac{1}{7} \\ - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array}]$

Step 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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Step 1.1

Consider the corresponding sign chart.

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

Step 1.2

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

Step 1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} \frac{1}{7} & - \frac{3}{7} & \frac{1}{14} \\ - \frac{2}{7} & \frac{11}{21} & \frac{1}{42} \\ \frac{6}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} |$

Step 1.4

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

$- \frac{4}{7} | \begin{array}{c} \frac{1}{7} & - \frac{3}{7} & \frac{1}{14} \\ - \frac{2}{7} & \frac{11}{21} & \frac{1}{42} \\ \frac{6}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} |$

Step 1.5

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

$| \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} |$

Step 1.6

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

$- \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} |$

Step 1.7

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

$| \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} |$

Step 1.8

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

$\frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} |$

Step 1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 1.10

Multiply element $a_{14}$ by its cofactor.

$- \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 1.11

Add the terms together.

$- \frac{4}{7} | \begin{array}{c} \frac{1}{7} & - \frac{3}{7} & \frac{1}{14} \\ - \frac{2}{7} & \frac{11}{21} & \frac{1}{42} \\ \frac{6}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2

Evaluate $| \begin{array}{c} \frac{1}{7} & - \frac{3}{7} & \frac{1}{14} \\ - \frac{2}{7} & \frac{11}{21} & \frac{1}{42} \\ \frac{6}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} |$ .

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Step 2.1

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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Step 2.1.1

Consider the corresponding sign chart.

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

Step 2.1.2

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

Step 2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} \frac{11}{21} & \frac{1}{42} \\ - \frac{18}{7} & - \frac{1}{14} \end{array} |$

Step 2.1.4

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

$\frac{1}{7} | \begin{array}{c} \frac{11}{21} & \frac{1}{42} \\ - \frac{18}{7} & - \frac{1}{14} \end{array} |$

Step 2.1.5

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

$| \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} |$

Step 2.1.6

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

$\frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} |$

Step 2.1.7

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

$| \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.1.8

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

$\frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.1.9

Add the terms together.

$- \frac{4}{7} (\frac{1}{7} | \begin{array}{c} \frac{11}{21} & \frac{1}{42} \\ - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2

Evaluate $| \begin{array}{c} \frac{11}{21} & \frac{1}{42} \\ - \frac{18}{7} & - \frac{1}{14} \end{array} |$ .

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Step 2.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- \frac{4}{7} (\frac{1}{7} (\frac{11}{21} (- \frac{1}{14}) - (- \frac{18}{7} \cdot \frac{1}{42})) + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2

Simplify the determinant.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $\frac{11}{21} (- \frac{1}{14})$ .

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Step 2.2.2.1.1.1

Multiply $\frac{11}{21}$ by $\frac{1}{14}$ .

$- \frac{4}{7} (\frac{1}{7} (- \frac{11}{21 \cdot 14} - (- \frac{18}{7} \cdot \frac{1}{42})) + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.1.1.2

Multiply $21$ by $14$ .

$- \frac{4}{7} (\frac{1}{7} (- \frac{11}{294} - (- \frac{18}{7} \cdot \frac{1}{42})) + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.1.2

Cancel the common factor of $6$ .

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Step 2.2.2.1.2.1

Move the leading negative in $- \frac{18}{7}$ into the numerator.

$- \frac{4}{7} (\frac{1}{7} (- \frac{11}{294} - (\frac{- 18}{7} \cdot \frac{1}{42})) + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.1.2.2

Factor $6$ out of $- 18$ .

$- \frac{4}{7} (\frac{1}{7} (- \frac{11}{294} - (\frac{6 (- 3)}{7} \cdot \frac{1}{42})) + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.1.2.3

Factor $6$ out of $42$ .

$- \frac{4}{7} (\frac{1}{7} (- \frac{11}{294} - (\frac{6 \cdot - 3}{7} \cdot \frac{1}{6 \cdot 7})) + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.1.2.4

Cancel the common factor.

Step 2.2.2.1.2.5

Rewrite the expression.

$- \frac{4}{7} (\frac{1}{7} (- \frac{11}{294} - (\frac{- 3}{7} \cdot \frac{1}{7})) + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.1.3

Multiply $\frac{- 3}{7}$ by $\frac{1}{7}$ .

$- \frac{4}{7} (\frac{1}{7} (- \frac{11}{294} - \frac{- 3}{7 \cdot 7}) + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.1.4

Multiply $7$ by $7$ .

$- \frac{4}{7} (\frac{1}{7} (- \frac{11}{294} - \frac{- 3}{49}) + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.1.5

Move the negative in front of the fraction.

$- \frac{4}{7} (\frac{1}{7} (- \frac{11}{294} - - \frac{3}{49}) + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.1.6

Multiply $- - \frac{3}{49}$ .

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Step 2.2.2.1.6.1

Multiply $- 1$ by $- 1$ .

$- \frac{4}{7} (\frac{1}{7} (- \frac{11}{294} + 1 (\frac{3}{49})) + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.1.6.2

Multiply $\frac{3}{49}$ by $1$ .

$- \frac{4}{7} (\frac{1}{7} (- \frac{11}{294} + \frac{3}{49}) + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.2

To write $\frac{3}{49}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{4}{7} (\frac{1}{7} (- \frac{11}{294} + \frac{3}{49} \cdot \frac{6}{6}) + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.3

Write each expression with a common denominator of $294$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.2.3.1

Multiply $\frac{3}{49}$ by $\frac{6}{6}$ .

$- \frac{4}{7} (\frac{1}{7} (- \frac{11}{294} + \frac{3 \cdot 6}{49 \cdot 6}) + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.3.2

Multiply $49$ by $6$ .

$- \frac{4}{7} (\frac{1}{7} (- \frac{11}{294} + \frac{3 \cdot 6}{294}) + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.4

Combine the numerators over the common denominator.

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{- 11 + 3 \cdot 6}{294} + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.5

Simplify the numerator.

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Step 2.2.2.5.1

Multiply $3$ by $6$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{- 11 + 18}{294} + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.5.2

Add $- 11$ and $18$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{7}{294} + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.6

Cancel the common factor of $7$ and $294$ .

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Step 2.2.2.6.1

Factor $7$ out of $7$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{7 (1)}{294} + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.6.2

Cancel the common factors.

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Step 2.2.2.6.2.1

Factor $7$ out of $294$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{7 \cdot 1}{7 \cdot 42} + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.2.2.6.2.2

Cancel the common factor.

Step 2.2.2.6.2.3

Rewrite the expression.

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3

Evaluate $| \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} |$ .

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Step 2.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} (- \frac{2}{7} (- \frac{1}{14}) - \frac{6}{7} \cdot \frac{1}{42}) + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2

Simplify the determinant.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1.1

Move the leading negative in $- \frac{2}{7}$ into the numerator.

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{- 2}{7} (- \frac{1}{14}) - \frac{6}{7} \cdot \frac{1}{42}) + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.1.1.2

Move the leading negative in $- \frac{1}{14}$ into the numerator.

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{- 2}{7} \cdot \frac{- 1}{14} - \frac{6}{7} \cdot \frac{1}{42}) + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.1.1.3

Factor $2$ out of $- 2$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{2 (- 1)}{7} \cdot \frac{- 1}{14} - \frac{6}{7} \cdot \frac{1}{42}) + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.1.1.4

Factor $2$ out of $14$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{2 \cdot - 1}{7} \cdot \frac{- 1}{2 \cdot 7} - \frac{6}{7} \cdot \frac{1}{42}) + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.1.1.5

Cancel the common factor.

Step 2.3.2.1.1.6

Rewrite the expression.

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{- 1}{7} \cdot \frac{- 1}{7} - \frac{6}{7} \cdot \frac{1}{42}) + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.1.2

Multiply $\frac{- 1}{7}$ by $\frac{- 1}{7}$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{- - 1}{7 \cdot 7} - \frac{6}{7} \cdot \frac{1}{42}) + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.1.3

Multiply $- 1$ by $- 1$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{1}{7 \cdot 7} - \frac{6}{7} \cdot \frac{1}{42}) + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.1.4

Multiply $7$ by $7$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{1}{49} - \frac{6}{7} \cdot \frac{1}{42}) + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.1.5

Cancel the common factor of $6$ .

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Step 2.3.2.1.5.1

Move the leading negative in $- \frac{6}{7}$ into the numerator.

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{1}{49} + \frac{- 6}{7} \cdot \frac{1}{42}) + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.1.5.2

Factor $6$ out of $- 6$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{1}{49} + \frac{6 (- 1)}{7} \cdot \frac{1}{42}) + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.1.5.3

Factor $6$ out of $42$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{1}{49} + \frac{6 \cdot - 1}{7} \cdot \frac{1}{6 \cdot 7}) + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.1.5.4

Cancel the common factor.

Step 2.3.2.1.5.5

Rewrite the expression.

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{1}{49} + \frac{- 1}{7} \cdot \frac{1}{7}) + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.1.6

Multiply $\frac{- 1}{7}$ by $\frac{1}{7}$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{1}{49} + \frac{- 1}{7 \cdot 7}) + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.1.7

Multiply $7$ by $7$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{1}{49} + \frac{- 1}{49}) + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.1.8

Move the negative in front of the fraction.

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{1}{49} - \frac{1}{49}) + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.2

Combine the numerators over the common denominator.

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1 - 1}{49} + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.3

Subtract $1$ from $1$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{0}{49} + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.3.2.4

Divide $0$ by $49$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4

Evaluate $| \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |$ .

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Step 2.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} (- \frac{2}{7} (- \frac{18}{7}) - \frac{6}{7} \cdot \frac{11}{21})) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2

Simplify the determinant.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Multiply $- \frac{2}{7} (- \frac{18}{7})$ .

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Step 2.4.2.1.1.1

Multiply $- 1$ by $- 1$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} (1 (\frac{2}{7}) \frac{18}{7} - \frac{6}{7} \cdot \frac{11}{21})) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.1.1.2

Multiply $\frac{2}{7}$ by $1$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} (\frac{2}{7} \cdot \frac{18}{7} - \frac{6}{7} \cdot \frac{11}{21})) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.1.1.3

Multiply $\frac{2}{7}$ by $\frac{18}{7}$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} (\frac{2 \cdot 18}{7 \cdot 7} - \frac{6}{7} \cdot \frac{11}{21})) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.1.1.4

Multiply $2$ by $18$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} (\frac{36}{7 \cdot 7} - \frac{6}{7} \cdot \frac{11}{21})) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.1.1.5

Multiply $7$ by $7$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} (\frac{36}{49} - \frac{6}{7} \cdot \frac{11}{21})) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.1.2

Cancel the common factor of $3$ .

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Step 2.4.2.1.2.1

Move the leading negative in $- \frac{6}{7}$ into the numerator.

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} (\frac{36}{49} + \frac{- 6}{7} \cdot \frac{11}{21})) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.1.2.2

Factor $3$ out of $- 6$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} (\frac{36}{49} + \frac{3 (- 2)}{7} \cdot \frac{11}{21})) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.1.2.3

Factor $3$ out of $21$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} (\frac{36}{49} + \frac{3 \cdot - 2}{7} \cdot \frac{11}{3 \cdot 7})) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.1.2.4

Cancel the common factor.

Step 2.4.2.1.2.5

Rewrite the expression.

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} (\frac{36}{49} + \frac{- 2}{7} \cdot \frac{11}{7})) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.1.3

Multiply $\frac{- 2}{7}$ by $\frac{11}{7}$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} (\frac{36}{49} + \frac{- 2 \cdot 11}{7 \cdot 7})) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.1.4

Multiply $- 2$ by $11$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} (\frac{36}{49} + \frac{- 22}{7 \cdot 7})) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.1.5

Multiply $7$ by $7$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} (\frac{36}{49} + \frac{- 22}{49})) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.1.6

Move the negative in front of the fraction.

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} (\frac{36}{49} - \frac{22}{49})) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.2

Combine the numerators over the common denominator.

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} \cdot \frac{36 - 22}{49}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.3

Subtract $22$ from $36$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} \cdot \frac{14}{49}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.4

Cancel the common factor of $14$ and $49$ .

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Step 2.4.2.4.1

Factor $7$ out of $14$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} \cdot \frac{7 (2)}{49}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.4.2

Cancel the common factors.

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Step 2.4.2.4.2.1

Factor $7$ out of $49$ .

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} \cdot \frac{7 \cdot 2}{7 \cdot 7}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.4.2.4.2.2

Cancel the common factor.

Step 2.4.2.4.2.3

Rewrite the expression.

$- \frac{4}{7} (\frac{1}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot 0 + \frac{1}{14} \cdot \frac{2}{7}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5

Simplify the determinant.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

Multiply $\frac{1}{7} \cdot \frac{1}{42}$ .

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Step 2.5.1.1.1

Multiply $\frac{1}{7}$ by $\frac{1}{42}$ .

$- \frac{4}{7} (\frac{1}{7 \cdot 42} + \frac{3}{7} \cdot 0 + \frac{1}{14} \cdot \frac{2}{7}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5.1.1.2

Multiply $7$ by $42$ .

$- \frac{4}{7} (\frac{1}{294} + \frac{3}{7} \cdot 0 + \frac{1}{14} \cdot \frac{2}{7}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5.1.2

Multiply $\frac{3}{7}$ by $0$ .

$- \frac{4}{7} (\frac{1}{294} + 0 + \frac{1}{14} \cdot \frac{2}{7}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5.1.3

Cancel the common factor of $2$ .

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Step 2.5.1.3.1

Factor $2$ out of $14$ .

$- \frac{4}{7} (\frac{1}{294} + 0 + \frac{1}{2 (7)} \cdot \frac{2}{7}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5.1.3.2

Cancel the common factor.

$- \frac{4}{7} (\frac{1}{294} + 0 + \frac{1}{2 \cdot 7} \cdot \frac{2}{7}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5.1.3.3

Rewrite the expression.

$- \frac{4}{7} (\frac{1}{294} + 0 + \frac{1}{7} \cdot \frac{1}{7}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5.1.4

Multiply $\frac{1}{7}$ by $\frac{1}{7}$ .

$- \frac{4}{7} (\frac{1}{294} + 0 + \frac{1}{7 \cdot 7}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5.1.5

Multiply $7$ by $7$ .

$- \frac{4}{7} (\frac{1}{294} + 0 + \frac{1}{49}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5.2

Add $\frac{1}{294}$ and $0$ .

$- \frac{4}{7} (\frac{1}{294} + \frac{1}{49}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5.3

To write $\frac{1}{49}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{4}{7} (\frac{1}{294} + \frac{1}{49} \cdot \frac{6}{6}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5.4

Write each expression with a common denominator of $294$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.4.1

Multiply $\frac{1}{49}$ by $\frac{6}{6}$ .

$- \frac{4}{7} (\frac{1}{294} + \frac{6}{49 \cdot 6}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5.4.2

Multiply $49$ by $6$ .

$- \frac{4}{7} (\frac{1}{294} + \frac{6}{294}) - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5.5

Combine the numerators over the common denominator.

$- \frac{4}{7} \cdot \frac{1 + 6}{294} - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5.6

Add $1$ and $6$ .

$- \frac{4}{7} \cdot \frac{7}{294} - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5.7

Cancel the common factor of $7$ and $294$ .

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Step 2.5.7.1

Factor $7$ out of $7$ .

$- \frac{4}{7} \cdot \frac{7 (1)}{294} - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5.7.2

Cancel the common factors.

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Step 2.5.7.2.1

Factor $7$ out of $294$ .

$- \frac{4}{7} \cdot \frac{7 \cdot 1}{7 \cdot 42} - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 2.5.7.2.2

Cancel the common factor.

Step 2.5.7.2.3

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} | \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3

Evaluate $| \begin{array}{c} - \frac{9}{7} & - \frac{3}{7} & \frac{1}{14} \\ \frac{40}{21} & \frac{11}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{18}{7} & - \frac{1}{14} \end{array} |$ .

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Step 3.1

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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Step 3.1.1

Consider the corresponding sign chart.

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

Step 3.1.2

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

Step 3.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} \frac{11}{21} & \frac{1}{42} \\ - \frac{18}{7} & - \frac{1}{14} \end{array} |$

Step 3.1.4

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

$- \frac{9}{7} | \begin{array}{c} \frac{11}{21} & \frac{1}{42} \\ - \frac{18}{7} & - \frac{1}{14} \end{array} |$

Step 3.1.5

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

$| \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} |$

Step 3.1.6

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

$\frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} |$

Step 3.1.7

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

$| \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |$

Step 3.1.8

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

$\frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |$

Step 3.1.9

Add the terms together.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} | \begin{array}{c} \frac{11}{21} & \frac{1}{42} \\ - \frac{18}{7} & - \frac{1}{14} \end{array} | + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2

Evaluate $| \begin{array}{c} \frac{11}{21} & \frac{1}{42} \\ - \frac{18}{7} & - \frac{1}{14} \end{array} |$ .

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Step 3.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} (\frac{11}{21} (- \frac{1}{14}) - (- \frac{18}{7} \cdot \frac{1}{42})) + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2

Simplify the determinant.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply $\frac{11}{21} (- \frac{1}{14})$ .

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Step 3.2.2.1.1.1

Multiply $\frac{11}{21}$ by $\frac{1}{14}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} (- \frac{11}{21 \cdot 14} - (- \frac{18}{7} \cdot \frac{1}{42})) + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.1.1.2

Multiply $21$ by $14$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} (- \frac{11}{294} - (- \frac{18}{7} \cdot \frac{1}{42})) + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.1.2

Cancel the common factor of $6$ .

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Step 3.2.2.1.2.1

Move the leading negative in $- \frac{18}{7}$ into the numerator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} (- \frac{11}{294} - (\frac{- 18}{7} \cdot \frac{1}{42})) + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.1.2.2

Factor $6$ out of $- 18$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} (- \frac{11}{294} - (\frac{6 (- 3)}{7} \cdot \frac{1}{42})) + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.1.2.3

Factor $6$ out of $42$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} (- \frac{11}{294} - (\frac{6 \cdot - 3}{7} \cdot \frac{1}{6 \cdot 7})) + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.1.2.4

Cancel the common factor.

Step 3.2.2.1.2.5

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} (- \frac{11}{294} - (\frac{- 3}{7} \cdot \frac{1}{7})) + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.1.3

Multiply $\frac{- 3}{7}$ by $\frac{1}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} (- \frac{11}{294} - \frac{- 3}{7 \cdot 7}) + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.1.4

Multiply $7$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} (- \frac{11}{294} - \frac{- 3}{49}) + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.1.5

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} (- \frac{11}{294} - - \frac{3}{49}) + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.1.6

Multiply $- - \frac{3}{49}$ .

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Step 3.2.2.1.6.1

Multiply $- 1$ by $- 1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} (- \frac{11}{294} + 1 (\frac{3}{49})) + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.1.6.2

Multiply $\frac{3}{49}$ by $1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} (- \frac{11}{294} + \frac{3}{49}) + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.2

To write $\frac{3}{49}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} (- \frac{11}{294} + \frac{3}{49} \cdot \frac{6}{6}) + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.3

Write each expression with a common denominator of $294$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.2.3.1

Multiply $\frac{3}{49}$ by $\frac{6}{6}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} (- \frac{11}{294} + \frac{3 \cdot 6}{49 \cdot 6}) + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.3.2

Multiply $49$ by $6$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} (- \frac{11}{294} + \frac{3 \cdot 6}{294}) + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.4

Combine the numerators over the common denominator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{- 11 + 3 \cdot 6}{294} + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.5

Simplify the numerator.

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Step 3.2.2.5.1

Multiply $3$ by $6$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{- 11 + 18}{294} + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.5.2

Add $- 11$ and $18$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{7}{294} + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.6

Cancel the common factor of $7$ and $294$ .

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Step 3.2.2.6.1

Factor $7$ out of $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{7 (1)}{294} + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.6.2

Cancel the common factors.

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Step 3.2.2.6.2.1

Factor $7$ out of $294$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{7 \cdot 1}{7 \cdot 42} + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.2.2.6.2.2

Cancel the common factor.

Step 3.2.2.6.2.3

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3

Evaluate $| \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} |$ .

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Step 3.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{40}{21} (- \frac{1}{14}) - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2

Simplify the determinant.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.1

Move the leading negative in $- \frac{1}{14}$ into the numerator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{40}{21} \cdot \frac{- 1}{14} - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.1.1.2

Factor $2$ out of $40$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{2 (20)}{21} \cdot \frac{- 1}{14} - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.1.1.3

Factor $2$ out of $14$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{2 \cdot 20}{21} \cdot \frac{- 1}{2 \cdot 7} - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.1.1.4

Cancel the common factor.

Step 3.3.2.1.1.5

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{20}{21} \cdot \frac{- 1}{7} - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.1.2

Multiply $\frac{20}{21}$ by $\frac{- 1}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{20 \cdot - 1}{21 \cdot 7} - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.1.3

Multiply $20$ by $- 1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{- 20}{21 \cdot 7} - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.1.4

Multiply $21$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} (\frac{- 20}{147} - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.1.5

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} (- \frac{20}{147} - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.1.6

Multiply $- \frac{47}{7} \cdot \frac{1}{42}$ .

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Step 3.3.2.1.6.1

Multiply $\frac{1}{42}$ by $\frac{47}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} (- \frac{20}{147} - - \frac{47}{42 \cdot 7}) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.1.6.2

Multiply $42$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} (- \frac{20}{147} - - \frac{47}{294}) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.1.7

Multiply $- - \frac{47}{294}$ .

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Step 3.3.2.1.7.1

Multiply $- 1$ by $- 1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} (- \frac{20}{147} + 1 (\frac{47}{294})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.1.7.2

Multiply $\frac{47}{294}$ by $1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} (- \frac{20}{147} + \frac{47}{294}) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.2

To write $- \frac{20}{147}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} (- \frac{20}{147} \cdot \frac{2}{2} + \frac{47}{294}) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.3

Write each expression with a common denominator of $294$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.2.3.1

Multiply $\frac{20}{147}$ by $\frac{2}{2}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} (- \frac{20 \cdot 2}{147 \cdot 2} + \frac{47}{294}) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.3.2

Multiply $147$ by $2$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} (- \frac{20 \cdot 2}{294} + \frac{47}{294}) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.4

Combine the numerators over the common denominator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{- 20 \cdot 2 + 47}{294} + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $- 20$ by $2$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{- 40 + 47}{294} + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.5.2

Add $- 40$ and $47$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{7}{294} + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.6

Cancel the common factor of $7$ and $294$ .

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Step 3.3.2.6.1

Factor $7$ out of $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{7 (1)}{294} + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.6.2

Cancel the common factors.

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Step 3.3.2.6.2.1

Factor $7$ out of $294$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{7 \cdot 1}{7 \cdot 42} + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.3.2.6.2.2

Cancel the common factor.

Step 3.3.2.6.2.3

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4

Evaluate $| \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |$ .

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Step 3.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{40}{21} (- \frac{18}{7}) - (- \frac{47}{7} \cdot \frac{11}{21}))) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2

Simplify the determinant.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Cancel the common factor of $3$ .

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Step 3.4.2.1.1.1

Move the leading negative in $- \frac{18}{7}$ into the numerator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{40}{21} \cdot \frac{- 18}{7} - (- \frac{47}{7} \cdot \frac{11}{21}))) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.1.1.2

Factor $3$ out of $21$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{40}{3 (7)} \cdot \frac{- 18}{7} - (- \frac{47}{7} \cdot \frac{11}{21}))) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.1.1.3

Factor $3$ out of $- 18$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{40}{3 \cdot 7} \cdot \frac{3 \cdot - 6}{7} - (- \frac{47}{7} \cdot \frac{11}{21}))) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.1.1.4

Cancel the common factor.

Step 3.4.2.1.1.5

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{40}{7} \cdot \frac{- 6}{7} - (- \frac{47}{7} \cdot \frac{11}{21}))) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.1.2

Multiply $\frac{40}{7}$ by $\frac{- 6}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{40 \cdot - 6}{7 \cdot 7} - (- \frac{47}{7} \cdot \frac{11}{21}))) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.1.3

Multiply $40$ by $- 6$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{- 240}{7 \cdot 7} - (- \frac{47}{7} \cdot \frac{11}{21}))) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.1.4

Multiply $7$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{- 240}{49} - (- \frac{47}{7} \cdot \frac{11}{21}))) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.1.5

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{240}{49} - (- \frac{47}{7} \cdot \frac{11}{21}))) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.1.6

Multiply $- \frac{47}{7} \cdot \frac{11}{21}$ .

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Step 3.4.2.1.6.1

Multiply $\frac{11}{21}$ by $\frac{47}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{240}{49} - - \frac{11 \cdot 47}{21 \cdot 7})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.1.6.2

Multiply $11$ by $47$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{240}{49} - - \frac{517}{21 \cdot 7})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.1.6.3

Multiply $21$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{240}{49} - - \frac{517}{147})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.1.7

Multiply $- - \frac{517}{147}$ .

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Step 3.4.2.1.7.1

Multiply $- 1$ by $- 1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{240}{49} + 1 (\frac{517}{147}))) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.1.7.2

Multiply $\frac{517}{147}$ by $1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{240}{49} + \frac{517}{147})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.2

To write $- \frac{240}{49}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{240}{49} \cdot \frac{3}{3} + \frac{517}{147})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.3

Write each expression with a common denominator of $147$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.2.3.1

Multiply $\frac{240}{49}$ by $\frac{3}{3}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{240 \cdot 3}{49 \cdot 3} + \frac{517}{147})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.3.2

Multiply $49$ by $3$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{240 \cdot 3}{147} + \frac{517}{147})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.4

Combine the numerators over the common denominator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} \cdot \frac{- 240 \cdot 3 + 517}{147}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.5

Simplify the numerator.

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Step 3.4.2.5.1

Multiply $- 240$ by $3$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} \cdot \frac{- 720 + 517}{147}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.5.2

Add $- 720$ and $517$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} \cdot \frac{- 203}{147}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.6

Cancel the common factor of $- 203$ and $147$ .

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Step 3.4.2.6.1

Factor $7$ out of $- 203$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} \cdot \frac{7 (- 29)}{147}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.6.2

Cancel the common factors.

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Step 3.4.2.6.2.1

Factor $7$ out of $147$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} \cdot \frac{7 \cdot - 29}{7 \cdot 21}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.6.2.2

Cancel the common factor.

Step 3.4.2.6.2.3

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} \cdot \frac{- 29}{21}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.4.2.7

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{29}{21})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5

Simplify the determinant.

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Step 3.5.1

Simplify each term.

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Step 3.5.1.1

Cancel the common factor of $3$ .

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Step 3.5.1.1.1

Move the leading negative in $- \frac{9}{7}$ into the numerator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (\frac{- 9}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{29}{21})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.1.1.2

Factor $3$ out of $- 9$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (\frac{3 (- 3)}{7} \cdot \frac{1}{42} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{29}{21})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.1.1.3

Factor $3$ out of $42$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (\frac{3 \cdot - 3}{7} \cdot \frac{1}{3 \cdot 14} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{29}{21})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.1.1.4

Cancel the common factor.

Step 3.5.1.1.5

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (\frac{- 3}{7} \cdot \frac{1}{14} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{29}{21})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.1.2

Multiply $\frac{- 3}{7}$ by $\frac{1}{14}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (\frac{- 3}{7 \cdot 14} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{29}{21})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.1.3

Multiply $7$ by $14$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (\frac{- 3}{98} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{29}{21})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.1.4

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{3}{98} + \frac{3}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{29}{21})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.1.5

Cancel the common factor of $3$ .

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Step 3.5.1.5.1

Factor $3$ out of $42$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{3}{98} + \frac{3}{7} \cdot \frac{1}{3 (14)} + \frac{1}{14} (- \frac{29}{21})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.1.5.2

Cancel the common factor.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{3}{98} + \frac{3}{7} \cdot \frac{1}{3 \cdot 14} + \frac{1}{14} (- \frac{29}{21})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.1.5.3

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{3}{98} + \frac{1}{7} \cdot \frac{1}{14} + \frac{1}{14} (- \frac{29}{21})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.1.6

Multiply $\frac{1}{7}$ by $\frac{1}{14}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{3}{98} + \frac{1}{7 \cdot 14} + \frac{1}{14} (- \frac{29}{21})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.1.7

Multiply $7$ by $14$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{3}{98} + \frac{1}{98} + \frac{1}{14} (- \frac{29}{21})) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.1.8

Multiply $\frac{1}{14} (- \frac{29}{21})$ .

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Step 3.5.1.8.1

Multiply $\frac{1}{14}$ by $\frac{29}{21}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{3}{98} + \frac{1}{98} - \frac{29}{14 \cdot 21}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.1.8.2

Multiply $14$ by $21$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{3}{98} + \frac{1}{98} - \frac{29}{294}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.2

Combine the numerators over the common denominator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (\frac{- 3 + 1}{98} + \frac{- 29}{294}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.3

Add $- 3$ and $1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (\frac{- 2}{98} + \frac{- 29}{294}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.4

Simplify each term.

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Step 3.5.4.1

Cancel the common factor of $- 2$ and $98$ .

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Step 3.5.4.1.1

Factor $2$ out of $- 2$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (\frac{2 (- 1)}{98} + \frac{- 29}{294}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.4.1.2

Cancel the common factors.

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Step 3.5.4.1.2.1

Factor $2$ out of $98$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (\frac{2 \cdot - 1}{2 \cdot 49} + \frac{- 29}{294}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.4.1.2.2

Cancel the common factor.

Step 3.5.4.1.2.3

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (\frac{- 1}{49} + \frac{- 29}{294}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.4.2

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{1}{49} + \frac{- 29}{294}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.4.3

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{1}{49} - \frac{29}{294}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.5

To write $- \frac{1}{49}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{1}{49} \cdot \frac{6}{6} - \frac{29}{294}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.6

Write each expression with a common denominator of $294$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.5.6.1

Multiply $\frac{1}{49}$ by $\frac{6}{6}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{6}{49 \cdot 6} - \frac{29}{294}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.6.2

Multiply $49$ by $6$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{6}{294} - \frac{29}{294}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.7

Combine the numerators over the common denominator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} \cdot \frac{- 6 - 29}{294} + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.8

Subtract $29$ from $- 6$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} \cdot \frac{- 35}{294} + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.9

Cancel the common factor of $- 35$ and $294$ .

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Step 3.5.9.1

Factor $7$ out of $- 35$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} \cdot \frac{7 (- 5)}{294} + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.9.2

Cancel the common factors.

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Step 3.5.9.2.1

Factor $7$ out of $294$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} \cdot \frac{7 \cdot - 5}{7 \cdot 42} + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.9.2.2

Cancel the common factor.

Step 3.5.9.2.3

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} \cdot \frac{- 5}{42} + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 3.5.10

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4

Evaluate $| \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & \frac{1}{14} \\ \frac{40}{21} & - \frac{2}{7} & \frac{1}{42} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{1}{14} \end{array} |$ .

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Step 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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Step 4.1.1

Consider the corresponding sign chart.

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

Step 4.1.2

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

Step 4.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} |$

Step 4.1.4

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

$- \frac{9}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} |$

Step 4.1.5

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

$| \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} |$

Step 4.1.6

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

$- \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} |$

Step 4.1.7

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

$| \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |$

Step 4.1.8

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

$\frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |$

Step 4.1.9

Add the terms together.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} | \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} | - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2

Evaluate $| \begin{array}{c} - \frac{2}{7} & \frac{1}{42} \\ \frac{6}{7} & - \frac{1}{14} \end{array} |$ .

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Step 4.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} (- \frac{2}{7} (- \frac{1}{14}) - \frac{6}{7} \cdot \frac{1}{42}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2

Simplify the determinant.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.1

Move the leading negative in $- \frac{2}{7}$ into the numerator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} (\frac{- 2}{7} (- \frac{1}{14}) - \frac{6}{7} \cdot \frac{1}{42}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.1.1.2

Move the leading negative in $- \frac{1}{14}$ into the numerator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} (\frac{- 2}{7} \cdot \frac{- 1}{14} - \frac{6}{7} \cdot \frac{1}{42}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.1.1.3

Factor $2$ out of $- 2$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} (\frac{2 (- 1)}{7} \cdot \frac{- 1}{14} - \frac{6}{7} \cdot \frac{1}{42}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.1.1.4

Factor $2$ out of $14$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} (\frac{2 \cdot - 1}{7} \cdot \frac{- 1}{2 \cdot 7} - \frac{6}{7} \cdot \frac{1}{42}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.1.1.5

Cancel the common factor.

Step 4.2.2.1.1.6

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} (\frac{- 1}{7} \cdot \frac{- 1}{7} - \frac{6}{7} \cdot \frac{1}{42}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.1.2

Multiply $\frac{- 1}{7}$ by $\frac{- 1}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} (\frac{- - 1}{7 \cdot 7} - \frac{6}{7} \cdot \frac{1}{42}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.1.3

Multiply $- 1$ by $- 1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} (\frac{1}{7 \cdot 7} - \frac{6}{7} \cdot \frac{1}{42}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.1.4

Multiply $7$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} (\frac{1}{49} - \frac{6}{7} \cdot \frac{1}{42}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.1.5

Cancel the common factor of $6$ .

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Step 4.2.2.1.5.1

Move the leading negative in $- \frac{6}{7}$ into the numerator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} (\frac{1}{49} + \frac{- 6}{7} \cdot \frac{1}{42}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.1.5.2

Factor $6$ out of $- 6$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} (\frac{1}{49} + \frac{6 (- 1)}{7} \cdot \frac{1}{42}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.1.5.3

Factor $6$ out of $42$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} (\frac{1}{49} + \frac{6 \cdot - 1}{7} \cdot \frac{1}{6 \cdot 7}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.1.5.4

Cancel the common factor.

Step 4.2.2.1.5.5

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} (\frac{1}{49} + \frac{- 1}{7} \cdot \frac{1}{7}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.1.6

Multiply $\frac{- 1}{7}$ by $\frac{1}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} (\frac{1}{49} + \frac{- 1}{7 \cdot 7}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.1.7

Multiply $7$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} (\frac{1}{49} + \frac{- 1}{49}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.1.8

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} (\frac{1}{49} - \frac{1}{49}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.2

Combine the numerators over the common denominator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot \frac{1 - 1}{49} - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.3

Subtract $1$ from $1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot \frac{0}{49} - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.2.2.4

Divide $0$ by $49$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} | + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3

Evaluate $| \begin{array}{c} \frac{40}{21} & \frac{1}{42} \\ - \frac{47}{7} & - \frac{1}{14} \end{array} |$ .

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Step 4.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} (\frac{40}{21} (- \frac{1}{14}) - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2

Simplify the determinant.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.1.1

Move the leading negative in $- \frac{1}{14}$ into the numerator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} (\frac{40}{21} \cdot \frac{- 1}{14} - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.1.1.2

Factor $2$ out of $40$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} (\frac{2 (20)}{21} \cdot \frac{- 1}{14} - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.1.1.3

Factor $2$ out of $14$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} (\frac{2 \cdot 20}{21} \cdot \frac{- 1}{2 \cdot 7} - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.1.1.4

Cancel the common factor.

Step 4.3.2.1.1.5

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} (\frac{20}{21} \cdot \frac{- 1}{7} - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.1.2

Multiply $\frac{20}{21}$ by $\frac{- 1}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} (\frac{20 \cdot - 1}{21 \cdot 7} - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.1.3

Multiply $20$ by $- 1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} (\frac{- 20}{21 \cdot 7} - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.1.4

Multiply $21$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} (\frac{- 20}{147} - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.1.5

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} (- \frac{20}{147} - (- \frac{47}{7} \cdot \frac{1}{42})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.1.6

Multiply $- \frac{47}{7} \cdot \frac{1}{42}$ .

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Step 4.3.2.1.6.1

Multiply $\frac{1}{42}$ by $\frac{47}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} (- \frac{20}{147} - - \frac{47}{42 \cdot 7}) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.1.6.2

Multiply $42$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} (- \frac{20}{147} - - \frac{47}{294}) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.1.7

Multiply $- - \frac{47}{294}$ .

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Step 4.3.2.1.7.1

Multiply $- 1$ by $- 1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} (- \frac{20}{147} + 1 (\frac{47}{294})) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.1.7.2

Multiply $\frac{47}{294}$ by $1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} (- \frac{20}{147} + \frac{47}{294}) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.2

To write $- \frac{20}{147}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} (- \frac{20}{147} \cdot \frac{2}{2} + \frac{47}{294}) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.3

Write each expression with a common denominator of $294$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.2.3.1

Multiply $\frac{20}{147}$ by $\frac{2}{2}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} (- \frac{20 \cdot 2}{147 \cdot 2} + \frac{47}{294}) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.3.2

Multiply $147$ by $2$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} (- \frac{20 \cdot 2}{294} + \frac{47}{294}) + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.4

Combine the numerators over the common denominator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{- 20 \cdot 2 + 47}{294} + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.5

Simplify the numerator.

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Step 4.3.2.5.1

Multiply $- 20$ by $2$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{- 40 + 47}{294} + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.5.2

Add $- 40$ and $47$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{7}{294} + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.6

Cancel the common factor of $7$ and $294$ .

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Step 4.3.2.6.1

Factor $7$ out of $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{7 (1)}{294} + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.6.2

Cancel the common factors.

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Step 4.3.2.6.2.1

Factor $7$ out of $294$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{7 \cdot 1}{7 \cdot 42} + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.3.2.6.2.2

Cancel the common factor.

Step 4.3.2.6.2.3

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4

Evaluate $| \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |$ .

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Step 4.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{40}{21} \cdot \frac{6}{7} - (- \frac{47}{7} (- \frac{2}{7})))) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2

Simplify the determinant.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

Cancel the common factor of $3$ .

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Step 4.4.2.1.1.1

Factor $3$ out of $21$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{40}{3 (7)} \cdot \frac{6}{7} - (- \frac{47}{7} (- \frac{2}{7})))) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2.1.1.2

Factor $3$ out of $6$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{40}{3 \cdot 7} \cdot \frac{3 \cdot 2}{7} - (- \frac{47}{7} (- \frac{2}{7})))) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2.1.1.3

Cancel the common factor.

Step 4.4.2.1.1.4

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{40}{7} \cdot \frac{2}{7} - (- \frac{47}{7} (- \frac{2}{7})))) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2.1.2

Multiply $\frac{40}{7}$ by $\frac{2}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{40 \cdot 2}{7 \cdot 7} - (- \frac{47}{7} (- \frac{2}{7})))) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2.1.3

Multiply $40$ by $2$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{80}{7 \cdot 7} - (- \frac{47}{7} (- \frac{2}{7})))) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2.1.4

Multiply $7$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{80}{49} - (- \frac{47}{7} (- \frac{2}{7})))) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2.1.5

Multiply $- \frac{47}{7} (- \frac{2}{7})$ .

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Step 4.4.2.1.5.1

Multiply $- 1$ by $- 1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{80}{49} - (1 (\frac{47}{7}) \frac{2}{7}))) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2.1.5.2

Multiply $\frac{47}{7}$ by $1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{80}{49} - (\frac{47}{7} \cdot \frac{2}{7}))) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2.1.5.3

Multiply $\frac{47}{7}$ by $\frac{2}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{80}{49} - \frac{47 \cdot 2}{7 \cdot 7})) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2.1.5.4

Multiply $47$ by $2$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{80}{49} - \frac{94}{7 \cdot 7})) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2.1.5.5

Multiply $7$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} (\frac{80}{49} - \frac{94}{49})) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2.2

Combine the numerators over the common denominator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} \cdot \frac{80 - 94}{49}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2.3

Subtract $94$ from $80$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} \cdot \frac{- 14}{49}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2.4

Cancel the common factor of $- 14$ and $49$ .

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Step 4.4.2.4.1

Factor $7$ out of $- 14$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} \cdot \frac{7 (- 2)}{49}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2.4.2

Cancel the common factors.

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Step 4.4.2.4.2.1

Factor $7$ out of $49$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} \cdot \frac{7 \cdot - 2}{7 \cdot 7}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2.4.2.2

Cancel the common factor.

Step 4.4.2.4.2.3

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} \cdot \frac{- 2}{7}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.4.2.5

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{9}{7} \cdot 0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{2}{7})) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5

Simplify the determinant.

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Step 4.5.1

Simplify each term.

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Step 4.5.1.1

Multiply $- \frac{9}{7} \cdot 0$ .

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Step 4.5.1.1.1

Multiply $0$ by $- 1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (0 (\frac{9}{7}) - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{2}{7})) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.1.1.2

Multiply $0$ by $\frac{9}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (0 - \frac{1}{7} \cdot \frac{1}{42} + \frac{1}{14} (- \frac{2}{7})) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.1.2

Multiply $- \frac{1}{7} \cdot \frac{1}{42}$ .

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Step 4.5.1.2.1

Multiply $\frac{1}{42}$ by $\frac{1}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (0 - \frac{1}{42 \cdot 7} + \frac{1}{14} (- \frac{2}{7})) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.1.2.2

Multiply $42$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (0 - \frac{1}{294} + \frac{1}{14} (- \frac{2}{7})) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.1.3

Cancel the common factor of $2$ .

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Step 4.5.1.3.1

Move the leading negative in $- \frac{2}{7}$ into the numerator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (0 - \frac{1}{294} + \frac{1}{14} \cdot \frac{- 2}{7}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.1.3.2

Factor $2$ out of $14$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (0 - \frac{1}{294} + \frac{1}{2 (7)} \cdot \frac{- 2}{7}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.1.3.3

Factor $2$ out of $- 2$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (0 - \frac{1}{294} + \frac{1}{2 \cdot 7} \cdot \frac{2 \cdot - 1}{7}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.1.3.4

Cancel the common factor.

Step 4.5.1.3.5

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (0 - \frac{1}{294} + \frac{1}{7} \cdot \frac{- 1}{7}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.1.4

Multiply $\frac{1}{7}$ by $\frac{- 1}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (0 - \frac{1}{294} + \frac{- 1}{7 \cdot 7}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.1.5

Multiply $7$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (0 - \frac{1}{294} + \frac{- 1}{49}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.1.6

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (0 - \frac{1}{294} - \frac{1}{49}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.2

Subtract $\frac{1}{294}$ from $0$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{294} - \frac{1}{49}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.3

To write $- \frac{1}{49}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{294} - \frac{1}{49} \cdot \frac{6}{6}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.4

Write each expression with a common denominator of $294$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.5.4.1

Multiply $\frac{1}{49}$ by $\frac{6}{6}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{294} - \frac{6}{49 \cdot 6}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.4.2

Multiply $49$ by $6$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{294} - \frac{6}{294}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.5

Combine the numerators over the common denominator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} \cdot \frac{- 1 - 6}{294} - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.6

Subtract $6$ from $- 1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} \cdot \frac{- 7}{294} - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.7

Cancel the common factor of $- 7$ and $294$ .

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Step 4.5.7.1

Factor $7$ out of $- 7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} \cdot \frac{7 (- 1)}{294} - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.7.2

Cancel the common factors.

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Step 4.5.7.2.1

Factor $7$ out of $294$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} \cdot \frac{7 \cdot - 1}{7 \cdot 42} - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.7.2.2

Cancel the common factor.

Step 4.5.7.2.3

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} \cdot \frac{- 1}{42} - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 4.5.8

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} | \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 5

Evaluate $| \begin{array}{c} - \frac{9}{7} & \frac{1}{7} & - \frac{3}{7} \\ \frac{40}{21} & - \frac{2}{7} & \frac{11}{21} \\ - \frac{47}{7} & \frac{6}{7} & - \frac{18}{7} \end{array} |$ .

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Step 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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Step 5.1.1

Consider the corresponding sign chart.

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

Step 5.1.2

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

Step 5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 5.1.4

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

$- \frac{9}{7} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |$

Step 5.1.5

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

$| \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |$

Step 5.1.6

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

$- \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |$

Step 5.1.7

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

$| \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |$

Step 5.1.8

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

$- \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |$

Step 5.1.9

Add the terms together.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} | \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} | - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2

Evaluate $| \begin{array}{c} - \frac{2}{7} & \frac{11}{21} \\ \frac{6}{7} & - \frac{18}{7} \end{array} |$ .

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Step 5.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} (- \frac{2}{7} (- \frac{18}{7}) - \frac{6}{7} \cdot \frac{11}{21}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2

Simplify the determinant.

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Step 5.2.2.1

Simplify each term.

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Step 5.2.2.1.1

Multiply $- \frac{2}{7} (- \frac{18}{7})$ .

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Step 5.2.2.1.1.1

Multiply $- 1$ by $- 1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} (1 (\frac{2}{7}) \frac{18}{7} - \frac{6}{7} \cdot \frac{11}{21}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.1.1.2

Multiply $\frac{2}{7}$ by $1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} (\frac{2}{7} \cdot \frac{18}{7} - \frac{6}{7} \cdot \frac{11}{21}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.1.1.3

Multiply $\frac{2}{7}$ by $\frac{18}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} (\frac{2 \cdot 18}{7 \cdot 7} - \frac{6}{7} \cdot \frac{11}{21}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.1.1.4

Multiply $2$ by $18$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} (\frac{36}{7 \cdot 7} - \frac{6}{7} \cdot \frac{11}{21}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.1.1.5

Multiply $7$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} (\frac{36}{49} - \frac{6}{7} \cdot \frac{11}{21}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.1.2

Cancel the common factor of $3$ .

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Step 5.2.2.1.2.1

Move the leading negative in $- \frac{6}{7}$ into the numerator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} (\frac{36}{49} + \frac{- 6}{7} \cdot \frac{11}{21}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.1.2.2

Factor $3$ out of $- 6$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} (\frac{36}{49} + \frac{3 (- 2)}{7} \cdot \frac{11}{21}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.1.2.3

Factor $3$ out of $21$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} (\frac{36}{49} + \frac{3 \cdot - 2}{7} \cdot \frac{11}{3 \cdot 7}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.1.2.4

Cancel the common factor.

Step 5.2.2.1.2.5

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} (\frac{36}{49} + \frac{- 2}{7} \cdot \frac{11}{7}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.1.3

Multiply $\frac{- 2}{7}$ by $\frac{11}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} (\frac{36}{49} + \frac{- 2 \cdot 11}{7 \cdot 7}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.1.4

Multiply $- 2$ by $11$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} (\frac{36}{49} + \frac{- 22}{7 \cdot 7}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.1.5

Multiply $7$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} (\frac{36}{49} + \frac{- 22}{49}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.1.6

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} (\frac{36}{49} - \frac{22}{49}) - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.2

Combine the numerators over the common denominator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{36 - 22}{49} - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.3

Subtract $22$ from $36$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{14}{49} - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.4

Cancel the common factor of $14$ and $49$ .

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Step 5.2.2.4.1

Factor $7$ out of $14$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{7 (2)}{49} - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.4.2

Cancel the common factors.

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Step 5.2.2.4.2.1

Factor $7$ out of $49$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{7 \cdot 2}{7 \cdot 7} - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.2.2.4.2.2

Cancel the common factor.

Step 5.2.2.4.2.3

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} | \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} | - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3

Evaluate $| \begin{array}{c} \frac{40}{21} & \frac{11}{21} \\ - \frac{47}{7} & - \frac{18}{7} \end{array} |$ .

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Step 5.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (\frac{40}{21} (- \frac{18}{7}) - (- \frac{47}{7} \cdot \frac{11}{21})) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2

Simplify the determinant.

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

Cancel the common factor of $3$ .

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Step 5.3.2.1.1.1

Move the leading negative in $- \frac{18}{7}$ into the numerator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (\frac{40}{21} \cdot \frac{- 18}{7} - (- \frac{47}{7} \cdot \frac{11}{21})) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.1.1.2

Factor $3$ out of $21$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (\frac{40}{3 (7)} \cdot \frac{- 18}{7} - (- \frac{47}{7} \cdot \frac{11}{21})) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.1.1.3

Factor $3$ out of $- 18$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (\frac{40}{3 \cdot 7} \cdot \frac{3 \cdot - 6}{7} - (- \frac{47}{7} \cdot \frac{11}{21})) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.1.1.4

Cancel the common factor.

Step 5.3.2.1.1.5

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (\frac{40}{7} \cdot \frac{- 6}{7} - (- \frac{47}{7} \cdot \frac{11}{21})) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.1.2

Multiply $\frac{40}{7}$ by $\frac{- 6}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (\frac{40 \cdot - 6}{7 \cdot 7} - (- \frac{47}{7} \cdot \frac{11}{21})) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.1.3

Multiply $40$ by $- 6$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (\frac{- 240}{7 \cdot 7} - (- \frac{47}{7} \cdot \frac{11}{21})) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.1.4

Multiply $7$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (\frac{- 240}{49} - (- \frac{47}{7} \cdot \frac{11}{21})) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.1.5

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{240}{49} - (- \frac{47}{7} \cdot \frac{11}{21})) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.1.6

Multiply $- \frac{47}{7} \cdot \frac{11}{21}$ .

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Step 5.3.2.1.6.1

Multiply $\frac{11}{21}$ by $\frac{47}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{240}{49} - - \frac{11 \cdot 47}{21 \cdot 7}) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.1.6.2

Multiply $11$ by $47$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{240}{49} - - \frac{517}{21 \cdot 7}) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.1.6.3

Multiply $21$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{240}{49} - - \frac{517}{147}) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.1.7

Multiply $- - \frac{517}{147}$ .

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Step 5.3.2.1.7.1

Multiply $- 1$ by $- 1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{240}{49} + 1 (\frac{517}{147})) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.1.7.2

Multiply $\frac{517}{147}$ by $1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{240}{49} + \frac{517}{147}) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.2

To write $- \frac{240}{49}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{240}{49} \cdot \frac{3}{3} + \frac{517}{147}) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.3

Write each expression with a common denominator of $147$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.2.3.1

Multiply $\frac{240}{49}$ by $\frac{3}{3}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{240 \cdot 3}{49 \cdot 3} + \frac{517}{147}) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.3.2

Multiply $49$ by $3$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{240 \cdot 3}{147} + \frac{517}{147}) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.4

Combine the numerators over the common denominator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} \cdot \frac{- 240 \cdot 3 + 517}{147} - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.5

Simplify the numerator.

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Step 5.3.2.5.1

Multiply $- 240$ by $3$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} \cdot \frac{- 720 + 517}{147} - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.5.2

Add $- 720$ and $517$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} \cdot \frac{- 203}{147} - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.6

Cancel the common factor of $- 203$ and $147$ .

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Step 5.3.2.6.1

Factor $7$ out of $- 203$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} \cdot \frac{7 (- 29)}{147} - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.6.2

Cancel the common factors.

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Step 5.3.2.6.2.1

Factor $7$ out of $147$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} \cdot \frac{7 \cdot - 29}{7 \cdot 21} - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.6.2.2

Cancel the common factor.

Step 5.3.2.6.2.3

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} \cdot \frac{- 29}{21} - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.3.2.7

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{29}{21}) - \frac{3}{7} | \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |)$

Step 5.4

Evaluate $| \begin{array}{c} \frac{40}{21} & - \frac{2}{7} \\ - \frac{47}{7} & \frac{6}{7} \end{array} |$ .

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Step 5.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{29}{21}) - \frac{3}{7} (\frac{40}{21} \cdot \frac{6}{7} - (- \frac{47}{7} (- \frac{2}{7}))))$

Step 5.4.2

Simplify the determinant.

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Step 5.4.2.1

Simplify each term.

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Step 5.4.2.1.1

Cancel the common factor of $3$ .

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Step 5.4.2.1.1.1

Factor $3$ out of $21$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{29}{21}) - \frac{3}{7} (\frac{40}{3 (7)} \cdot \frac{6}{7} - (- \frac{47}{7} (- \frac{2}{7}))))$

Step 5.4.2.1.1.2

Factor $3$ out of $6$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{29}{21}) - \frac{3}{7} (\frac{40}{3 \cdot 7} \cdot \frac{3 \cdot 2}{7} - (- \frac{47}{7} (- \frac{2}{7}))))$

Step 5.4.2.1.1.3

Cancel the common factor.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{29}{21}) - \frac{3}{7} (\frac{40}{3 \cdot 7} \cdot \frac{3 \cdot 2}{7} - (- \frac{47}{7} (- \frac{2}{7}))))$

Step 5.4.2.1.1.4

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{29}{21}) - \frac{3}{7} (\frac{40}{7} \cdot \frac{2}{7} - (- \frac{47}{7} (- \frac{2}{7}))))$

Step 5.4.2.1.2

Multiply $\frac{40}{7}$ by $\frac{2}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{29}{21}) - \frac{3}{7} (\frac{40 \cdot 2}{7 \cdot 7} - (- \frac{47}{7} (- \frac{2}{7}))))$

Step 5.4.2.1.3

Multiply $40$ by $2$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{29}{21}) - \frac{3}{7} (\frac{80}{7 \cdot 7} - (- \frac{47}{7} (- \frac{2}{7}))))$

Step 5.4.2.1.4

Multiply $7$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{29}{21}) - \frac{3}{7} (\frac{80}{49} - (- \frac{47}{7} (- \frac{2}{7}))))$

Step 5.4.2.1.5

Multiply $- \frac{47}{7} (- \frac{2}{7})$ .

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Step 5.4.2.1.5.1

Multiply $- 1$ by $- 1$ .

Step 5.4.2.1.5.2

Multiply $\frac{47}{7}$ by $1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{9}{7} \cdot \frac{2}{7} - \frac{1}{7} (- \frac{29}{21}) - \frac{3}{7} (\frac{80}{49} - (\frac{47}{7} \cdot \frac{2}{7})))$

Step 5.4.2.1.5.3

Multiply $\frac{47}{7}$ by $\frac{2}{7}$ .

Step 5.4.2.1.5.4

Multiply $47$ by $2$ .

Step 5.4.2.1.5.5

Multiply $7$ by $7$ .

Step 5.4.2.2

Combine the numerators over the common denominator.

Step 5.4.2.3

Subtract $94$ from $80$ .

Step 5.4.2.4

Cancel the common factor of $- 14$ and $49$ .

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Step 5.4.2.4.1

Factor $7$ out of $- 14$ .

Step 5.4.2.4.2

Cancel the common factors.

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Step 5.4.2.4.2.1

Factor $7$ out of $49$ .

Step 5.4.2.4.2.2

Cancel the common factor.

Step 5.4.2.4.2.3

Rewrite the expression.

Step 5.4.2.5

Move the negative in front of the fraction.

Step 5.5

Simplify the determinant.

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Step 5.5.1

Simplify each term.

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Step 5.5.1.1

Multiply $- \frac{9}{7} \cdot \frac{2}{7}$ .

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Step 5.5.1.1.1

Multiply $\frac{2}{7}$ by $\frac{9}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{2 \cdot 9}{7 \cdot 7} - \frac{1}{7} (- \frac{29}{21}) - \frac{3}{7} (- \frac{2}{7}))$

Step 5.5.1.1.2

Multiply $2$ by $9$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{18}{7 \cdot 7} - \frac{1}{7} (- \frac{29}{21}) - \frac{3}{7} (- \frac{2}{7}))$

Step 5.5.1.1.3

Multiply $7$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{18}{49} - \frac{1}{7} (- \frac{29}{21}) - \frac{3}{7} (- \frac{2}{7}))$

Step 5.5.1.2

Multiply $- \frac{1}{7} (- \frac{29}{21})$ .

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Step 5.5.1.2.1

Multiply $- 1$ by $- 1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{18}{49} + 1 (\frac{1}{7}) \frac{29}{21} - \frac{3}{7} (- \frac{2}{7}))$

Step 5.5.1.2.2

Multiply $\frac{1}{7}$ by $1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{18}{49} + \frac{1}{7} \cdot \frac{29}{21} - \frac{3}{7} (- \frac{2}{7}))$

Step 5.5.1.2.3

Multiply $\frac{1}{7}$ by $\frac{29}{21}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{18}{49} + \frac{29}{7 \cdot 21} - \frac{3}{7} (- \frac{2}{7}))$

Step 5.5.1.2.4

Multiply $7$ by $21$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{18}{49} + \frac{29}{147} - \frac{3}{7} (- \frac{2}{7}))$

Step 5.5.1.3

Multiply $- \frac{3}{7} (- \frac{2}{7})$ .

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Step 5.5.1.3.1

Multiply $- 1$ by $- 1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{18}{49} + \frac{29}{147} + 1 (\frac{3}{7}) \frac{2}{7})$

Step 5.5.1.3.2

Multiply $\frac{3}{7}$ by $1$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{18}{49} + \frac{29}{147} + \frac{3}{7} \cdot \frac{2}{7})$

Step 5.5.1.3.3

Multiply $\frac{3}{7}$ by $\frac{2}{7}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{18}{49} + \frac{29}{147} + \frac{3 \cdot 2}{7 \cdot 7})$

Step 5.5.1.3.4

Multiply $3$ by $2$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{18}{49} + \frac{29}{147} + \frac{6}{7 \cdot 7})$

Step 5.5.1.3.5

Multiply $7$ by $7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{18}{49} + \frac{29}{147} + \frac{6}{49})$

Step 5.5.2

Combine the numerators over the common denominator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (\frac{- 18 + 6}{49} + \frac{29}{147})$

Step 5.5.3

Add $- 18$ and $6$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (\frac{- 12}{49} + \frac{29}{147})$

Step 5.5.4

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{12}{49} + \frac{29}{147})$

Step 5.5.5

To write $- \frac{12}{49}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{12}{49} \cdot \frac{3}{3} + \frac{29}{147})$

Step 5.5.6

Write each expression with a common denominator of $147$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.6.1

Multiply $\frac{12}{49}$ by $\frac{3}{3}$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{12 \cdot 3}{49 \cdot 3} + \frac{29}{147})$

Step 5.5.6.2

Multiply $49$ by $3$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{12 \cdot 3}{147} + \frac{29}{147})$

Step 5.5.7

Combine the numerators over the common denominator.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} \cdot \frac{- 12 \cdot 3 + 29}{147}$

Step 5.5.8

Simplify the numerator.

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Step 5.5.8.1

Multiply $- 12$ by $3$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} \cdot \frac{- 36 + 29}{147}$

Step 5.5.8.2

Add $- 36$ and $29$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} \cdot \frac{- 7}{147}$

Step 5.5.9

Cancel the common factor of $- 7$ and $147$ .

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Step 5.5.9.1

Factor $7$ out of $- 7$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} \cdot \frac{7 (- 1)}{147}$

Step 5.5.9.2

Cancel the common factors.

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Step 5.5.9.2.1

Factor $7$ out of $147$ .

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} \cdot \frac{7 \cdot - 1}{7 \cdot 21}$

Step 5.5.9.2.2

Cancel the common factor.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} \cdot \frac{7 \cdot - 1}{7 \cdot 21}$

Step 5.5.9.2.3

Rewrite the expression.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} \cdot \frac{- 1}{21}$

Step 5.5.10

Move the negative in front of the fraction.

$- \frac{4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6

Simplify the determinant.

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Step 6.1

Simplify each term.

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Step 6.1.1

Cancel the common factor of $2$ .

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Step 6.1.1.1

Move the leading negative in $- \frac{4}{7}$ into the numerator.

$\frac{- 4}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.1.2

Factor $2$ out of $- 4$ .

$\frac{2 (- 2)}{7} \cdot \frac{1}{42} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.1.3

Factor $2$ out of $42$ .

$\frac{2 \cdot - 2}{7} \cdot \frac{1}{2 \cdot 21} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.1.4

Cancel the common factor.

$\frac{2 \cdot - 2}{7} \cdot \frac{1}{2 \cdot 21} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.1.5

Rewrite the expression.

$\frac{- 2}{7} \cdot \frac{1}{21} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.2

Multiply $\frac{- 2}{7}$ by $\frac{1}{21}$ .

$\frac{- 2}{7 \cdot 21} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.3

Multiply $7$ by $21$ .

$\frac{- 2}{147} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.4

Move the negative in front of the fraction.

$- \frac{2}{147} - \frac{2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.5

Cancel the common factor of $2$ .

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Step 6.1.5.1

Move the leading negative in $- \frac{2}{7}$ into the numerator.

$- \frac{2}{147} + \frac{- 2}{7} (- \frac{5}{42}) + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.5.2

Move the leading negative in $- \frac{5}{42}$ into the numerator.

$- \frac{2}{147} + \frac{- 2}{7} \cdot \frac{- 5}{42} + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.5.3

Factor $2$ out of $- 2$ .

$- \frac{2}{147} + \frac{2 (- 1)}{7} \cdot \frac{- 5}{42} + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.5.4

Factor $2$ out of $42$ .

$- \frac{2}{147} + \frac{2 \cdot - 1}{7} \cdot \frac{- 5}{2 \cdot 21} + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.5.5

Cancel the common factor.

$- \frac{2}{147} + \frac{2 \cdot - 1}{7} \cdot \frac{- 5}{2 \cdot 21} + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.5.6

Rewrite the expression.

$- \frac{2}{147} + \frac{- 1}{7} \cdot \frac{- 5}{21} + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.6

Multiply $\frac{- 1}{7}$ by $\frac{- 5}{21}$ .

$- \frac{2}{147} + \frac{- - 5}{7 \cdot 21} + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.7

Multiply $- 1$ by $- 5$ .

$- \frac{2}{147} + \frac{5}{7 \cdot 21} + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.8

Multiply $7$ by $21$ .

$- \frac{2}{147} + \frac{5}{147} + \frac{1}{7} (- \frac{1}{42}) - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.9

Multiply $\frac{1}{7} (- \frac{1}{42})$ .

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Step 6.1.9.1

Multiply $\frac{1}{7}$ by $\frac{1}{42}$ .

$- \frac{2}{147} + \frac{5}{147} - \frac{1}{7 \cdot 42} - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.9.2

Multiply $7$ by $42$ .

$- \frac{2}{147} + \frac{5}{147} - \frac{1}{294} - \frac{1}{7} (- \frac{1}{21})$

Step 6.1.10

Multiply $- \frac{1}{7} (- \frac{1}{21})$ .

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Step 6.1.10.1

Multiply $- 1$ by $- 1$ .

$- \frac{2}{147} + \frac{5}{147} - \frac{1}{294} + 1 (\frac{1}{7}) \frac{1}{21}$

Step 6.1.10.2

Multiply $\frac{1}{7}$ by $1$ .

$- \frac{2}{147} + \frac{5}{147} - \frac{1}{294} + \frac{1}{7} \cdot \frac{1}{21}$

Step 6.1.10.3

Multiply $\frac{1}{7}$ by $\frac{1}{21}$ .

$- \frac{2}{147} + \frac{5}{147} - \frac{1}{294} + \frac{1}{7 \cdot 21}$

Step 6.1.10.4

Multiply $7$ by $21$ .

$- \frac{2}{147} + \frac{5}{147} - \frac{1}{294} + \frac{1}{147}$

Step 6.2

Combine the numerators over the common denominator.

$\frac{- 2 + 5 + 1}{147} + \frac{- 1}{294}$

Step 6.3

Add $- 2$ and $5$ .

$\frac{3 + 1}{147} + \frac{- 1}{294}$

Step 6.4

Add $3$ and $1$ .

$\frac{4}{147} + \frac{- 1}{294}$

Step 6.5

Move the negative in front of the fraction.

$\frac{4}{147} - \frac{1}{294}$

Step 6.6

To write $\frac{4}{147}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{4}{147} \cdot \frac{2}{2} - \frac{1}{294}$

Step 6.7

Write each expression with a common denominator of $294$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.7.1

Multiply $\frac{4}{147}$ by $\frac{2}{2}$ .

$\frac{4 \cdot 2}{147 \cdot 2} - \frac{1}{294}$

Step 6.7.2

Multiply $147$ by $2$ .

$\frac{4 \cdot 2}{294} - \frac{1}{294}$

Step 6.8

Combine the numerators over the common denominator.

$\frac{4 \cdot 2 - 1}{294}$

Step 6.9

Simplify the numerator.

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Step 6.9.1

Multiply $4$ by $2$ .

$\frac{8 - 1}{294}$

Step 6.9.2

Subtract $1$ from $8$ .

$\frac{7}{294}$

Step 6.10

Cancel the common factor of $7$ and $294$ .

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Step 6.10.1

Factor $7$ out of $7$ .

$\frac{7 (1)}{294}$

Step 6.10.2

Cancel the common factors.

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Step 6.10.2.1

Factor $7$ out of $294$ .

$\frac{7 \cdot 1}{7 \cdot 42}$

Step 6.10.2.2

Cancel the common factor.

$\frac{7 \cdot 1}{7 \cdot 42}$

Step 6.10.2.3

Rewrite the expression.

$\frac{1}{42}$