Linear Algebra Examples

$[\begin{array}{c} \frac{17}{26} & \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{12}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{17}{26} & \frac{16}{26} \end{array}]$

Step 1

Cancel the common factor of $12$ and $26$ .

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Step 1.1

Factor $2$ out of $12$ .

$[\begin{array}{c} \frac{17}{26} & \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{2 (6)}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{17}{26} & \frac{16}{26} \end{array}]$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $26$ .

$[\begin{array}{c} \frac{17}{26} & \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{2 \cdot 6}{2 \cdot 13} & \frac{17}{26} \\ \frac{17}{26} & \frac{17}{26} & \frac{16}{26} \end{array}]$

Step 1.2.2

Cancel the common factor.

$[\begin{array}{c} \frac{17}{26} & \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{2 \cdot 6}{2 \cdot 13} & \frac{17}{26} \\ \frac{17}{26} & \frac{17}{26} & \frac{16}{26} \end{array}]$

Step 1.2.3

Rewrite the expression.

$[\begin{array}{c} \frac{17}{26} & \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{6}{13} & \frac{17}{26} \\ \frac{17}{26} & \frac{17}{26} & \frac{16}{26} \end{array}]$

Step 2

Cancel the common factor of $16$ and $26$ .

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Step 2.1

Factor $2$ out of $16$ .

$[\begin{array}{c} \frac{17}{26} & \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{6}{13} & \frac{17}{26} \\ \frac{17}{26} & \frac{17}{26} & \frac{2 (8)}{26} \end{array}]$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $2$ out of $26$ .

$[\begin{array}{c} \frac{17}{26} & \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{6}{13} & \frac{17}{26} \\ \frac{17}{26} & \frac{17}{26} & \frac{2 \cdot 8}{2 \cdot 13} \end{array}]$

Step 2.2.2

Cancel the common factor.

$[\begin{array}{c} \frac{17}{26} & \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{6}{13} & \frac{17}{26} \\ \frac{17}{26} & \frac{17}{26} & \frac{2 \cdot 8}{2 \cdot 13} \end{array}]$

Step 2.2.3

Rewrite the expression.

$[\begin{array}{c} \frac{17}{26} & \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{6}{13} & \frac{17}{26} \\ \frac{17}{26} & \frac{17}{26} & \frac{8}{13} \end{array}]$

Step 3

Find the determinant.

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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.

$| \begin{array}{c} \frac{6}{13} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} |$

Step 3.1.4

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

$\frac{17}{26} | \begin{array}{c} \frac{6}{13} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} |$

Step 3.1.5

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

$| \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} |$

Step 3.1.6

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

$- \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} |$

Step 3.1.7

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

$| \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.1.8

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

$\frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.1.9

Add the terms together.

$\frac{17}{26} | \begin{array}{c} \frac{6}{13} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} | - \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} | + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.2

Evaluate $| \begin{array}{c} \frac{6}{13} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} |$ .

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Step 3.2.1

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

$\frac{17}{26} (\frac{6}{13} \cdot \frac{8}{13} - \frac{17}{26} \cdot \frac{17}{26}) - \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} | + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

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 $\frac{6}{13} \cdot \frac{8}{13}$ .

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Step 3.2.2.1.1.1

Multiply $\frac{6}{13}$ by $\frac{8}{13}$ .

$\frac{17}{26} (\frac{6 \cdot 8}{13 \cdot 13} - \frac{17}{26} \cdot \frac{17}{26}) - \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} | + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.2.2.1.1.2

Multiply $6$ by $8$ .

$\frac{17}{26} (\frac{48}{13 \cdot 13} - \frac{17}{26} \cdot \frac{17}{26}) - \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} | + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.2.2.1.1.3

Multiply $13$ by $13$ .

$\frac{17}{26} (\frac{48}{169} - \frac{17}{26} \cdot \frac{17}{26}) - \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} | + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.2.2.1.2

Multiply $- \frac{17}{26} \cdot \frac{17}{26}$ .

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Step 3.2.2.1.2.1

Multiply $\frac{17}{26}$ by $\frac{17}{26}$ .

$\frac{17}{26} (\frac{48}{169} - \frac{17 \cdot 17}{26 \cdot 26}) - \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} | + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.2.2.1.2.2

Multiply $17$ by $17$ .

$\frac{17}{26} (\frac{48}{169} - \frac{289}{26 \cdot 26}) - \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} | + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.2.2.1.2.3

Multiply $26$ by $26$ .

$\frac{17}{26} (\frac{48}{169} - \frac{289}{676}) - \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} | + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.2.2.2

To write $\frac{48}{169}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{17}{26} (\frac{48}{169} \cdot \frac{4}{4} - \frac{289}{676}) - \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} | + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.2.2.3

Write each expression with a common denominator of $676$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.2.3.1

Multiply $\frac{48}{169}$ by $\frac{4}{4}$ .

$\frac{17}{26} (\frac{48 \cdot 4}{169 \cdot 4} - \frac{289}{676}) - \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} | + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.2.2.3.2

Multiply $169$ by $4$ .

$\frac{17}{26} (\frac{48 \cdot 4}{676} - \frac{289}{676}) - \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} | + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.2.2.4

Combine the numerators over the common denominator.

$\frac{17}{26} \cdot \frac{48 \cdot 4 - 289}{676} - \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} | + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.2.2.5

Simplify the numerator.

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Step 3.2.2.5.1

Multiply $48$ by $4$ .

$\frac{17}{26} \cdot \frac{192 - 289}{676} - \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} | + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.2.2.5.2

Subtract $289$ from $192$ .

$\frac{17}{26} \cdot \frac{- 97}{676} - \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} | + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.2.2.6

Move the negative in front of the fraction.

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} | + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.3

Evaluate $| \begin{array}{c} \frac{17}{26} & \frac{17}{26} \\ \frac{17}{26} & \frac{8}{13} \end{array} |$ .

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Step 3.3.1

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

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (\frac{17}{26} \cdot \frac{8}{13} - \frac{17}{26} \cdot \frac{17}{26}) + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

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

Factor $2$ out of $26$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (\frac{17}{2 (13)} \cdot \frac{8}{13} - \frac{17}{26} \cdot \frac{17}{26}) + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.3.2.1.1.2

Factor $2$ out of $8$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (\frac{17}{2 \cdot 13} \cdot \frac{2 \cdot 4}{13} - \frac{17}{26} \cdot \frac{17}{26}) + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.3.2.1.1.3

Cancel the common factor.

Step 3.3.2.1.1.4

Rewrite the expression.

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (\frac{17}{13} \cdot \frac{4}{13} - \frac{17}{26} \cdot \frac{17}{26}) + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.3.2.1.2

Multiply $\frac{17}{13}$ by $\frac{4}{13}$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (\frac{17 \cdot 4}{13 \cdot 13} - \frac{17}{26} \cdot \frac{17}{26}) + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.3.2.1.3

Multiply $17$ by $4$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (\frac{68}{13 \cdot 13} - \frac{17}{26} \cdot \frac{17}{26}) + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.3.2.1.4

Multiply $13$ by $13$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (\frac{68}{169} - \frac{17}{26} \cdot \frac{17}{26}) + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.3.2.1.5

Multiply $- \frac{17}{26} \cdot \frac{17}{26}$ .

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Step 3.3.2.1.5.1

Multiply $\frac{17}{26}$ by $\frac{17}{26}$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (\frac{68}{169} - \frac{17 \cdot 17}{26 \cdot 26}) + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.3.2.1.5.2

Multiply $17$ by $17$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (\frac{68}{169} - \frac{289}{26 \cdot 26}) + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.3.2.1.5.3

Multiply $26$ by $26$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (\frac{68}{169} - \frac{289}{676}) + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.3.2.2

To write $\frac{68}{169}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (\frac{68}{169} \cdot \frac{4}{4} - \frac{289}{676}) + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.3.2.3

Write each expression with a common denominator of $676$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.2.3.1

Multiply $\frac{68}{169}$ by $\frac{4}{4}$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (\frac{68 \cdot 4}{169 \cdot 4} - \frac{289}{676}) + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.3.2.3.2

Multiply $169$ by $4$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (\frac{68 \cdot 4}{676} - \frac{289}{676}) + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.3.2.4

Combine the numerators over the common denominator.

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} \cdot \frac{68 \cdot 4 - 289}{676} + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $68$ by $4$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} \cdot \frac{272 - 289}{676} + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.3.2.5.2

Subtract $289$ from $272$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} \cdot \frac{- 17}{676} + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.3.2.6

Move the negative in front of the fraction.

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} | \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$

Step 3.4

Evaluate $| \begin{array}{c} \frac{17}{26} & \frac{6}{13} \\ \frac{17}{26} & \frac{17}{26} \end{array} |$ .

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Step 3.4.1

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

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} (\frac{17}{26} \cdot \frac{17}{26} - \frac{17}{26} \cdot \frac{6}{13})$

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

Multiply $\frac{17}{26} \cdot \frac{17}{26}$ .

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Step 3.4.2.1.1.1

Multiply $\frac{17}{26}$ by $\frac{17}{26}$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} (\frac{17 \cdot 17}{26 \cdot 26} - \frac{17}{26} \cdot \frac{6}{13})$

Step 3.4.2.1.1.2

Multiply $17$ by $17$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} (\frac{289}{26 \cdot 26} - \frac{17}{26} \cdot \frac{6}{13})$

Step 3.4.2.1.1.3

Multiply $26$ by $26$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} (\frac{289}{676} - \frac{17}{26} \cdot \frac{6}{13})$

Step 3.4.2.1.2

Cancel the common factor of $2$ .

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Step 3.4.2.1.2.1

Move the leading negative in $- \frac{17}{26}$ into the numerator.

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} (\frac{289}{676} + \frac{- 17}{26} \cdot \frac{6}{13})$

Step 3.4.2.1.2.2

Factor $2$ out of $26$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} (\frac{289}{676} + \frac{- 17}{2 (13)} \cdot \frac{6}{13})$

Step 3.4.2.1.2.3

Factor $2$ out of $6$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} (\frac{289}{676} + \frac{- 17}{2 \cdot 13} \cdot \frac{2 \cdot 3}{13})$

Step 3.4.2.1.2.4

Cancel the common factor.

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} (\frac{289}{676} + \frac{- 17}{2 \cdot 13} \cdot \frac{2 \cdot 3}{13})$

Step 3.4.2.1.2.5

Rewrite the expression.

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} (\frac{289}{676} + \frac{- 17}{13} \cdot \frac{3}{13})$

Step 3.4.2.1.3

Multiply $\frac{- 17}{13}$ by $\frac{3}{13}$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} (\frac{289}{676} + \frac{- 17 \cdot 3}{13 \cdot 13})$

Step 3.4.2.1.4

Multiply $- 17$ by $3$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} (\frac{289}{676} + \frac{- 51}{13 \cdot 13})$

Step 3.4.2.1.5

Multiply $13$ by $13$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} (\frac{289}{676} + \frac{- 51}{169})$

Step 3.4.2.1.6

Move the negative in front of the fraction.

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} (\frac{289}{676} - \frac{51}{169})$

Step 3.4.2.2

To write $- \frac{51}{169}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} (\frac{289}{676} - \frac{51}{169} \cdot \frac{4}{4})$

Step 3.4.2.3

Write each expression with a common denominator of $676$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.2.3.1

Multiply $\frac{51}{169}$ by $\frac{4}{4}$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} (\frac{289}{676} - \frac{51 \cdot 4}{169 \cdot 4})$

Step 3.4.2.3.2

Multiply $169$ by $4$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} (\frac{289}{676} - \frac{51 \cdot 4}{676})$

Step 3.4.2.4

Combine the numerators over the common denominator.

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} \cdot \frac{289 - 51 \cdot 4}{676}$

Step 3.4.2.5

Simplify the numerator.

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Step 3.4.2.5.1

Multiply $- 51$ by $4$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} \cdot \frac{289 - 204}{676}$

Step 3.4.2.5.2

Subtract $204$ from $289$ .

$\frac{17}{26} (- \frac{97}{676}) - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} \cdot \frac{85}{676}$

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

Multiply $\frac{17}{26} (- \frac{97}{676})$ .

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Step 3.5.1.1.1

Multiply $\frac{17}{26}$ by $\frac{97}{676}$ .

$- \frac{17 \cdot 97}{26 \cdot 676} - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} \cdot \frac{85}{676}$

Step 3.5.1.1.2

Multiply $17$ by $97$ .

$- \frac{1649}{26 \cdot 676} - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} \cdot \frac{85}{676}$

Step 3.5.1.1.3

Multiply $26$ by $676$ .

$- \frac{1649}{17576} - \frac{17}{26} (- \frac{17}{676}) + \frac{17}{26} \cdot \frac{85}{676}$

Step 3.5.1.2

Multiply $- \frac{17}{26} (- \frac{17}{676})$ .

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Step 3.5.1.2.1

Multiply $- 1$ by $- 1$ .

$- \frac{1649}{17576} + 1 (\frac{17}{26}) \frac{17}{676} + \frac{17}{26} \cdot \frac{85}{676}$

Step 3.5.1.2.2

Multiply $\frac{17}{26}$ by $1$ .

$- \frac{1649}{17576} + \frac{17}{26} \cdot \frac{17}{676} + \frac{17}{26} \cdot \frac{85}{676}$

Step 3.5.1.2.3

Multiply $\frac{17}{26}$ by $\frac{17}{676}$ .

$- \frac{1649}{17576} + \frac{17 \cdot 17}{26 \cdot 676} + \frac{17}{26} \cdot \frac{85}{676}$

Step 3.5.1.2.4

Multiply $17$ by $17$ .

$- \frac{1649}{17576} + \frac{289}{26 \cdot 676} + \frac{17}{26} \cdot \frac{85}{676}$

Step 3.5.1.2.5

Multiply $26$ by $676$ .

$- \frac{1649}{17576} + \frac{289}{17576} + \frac{17}{26} \cdot \frac{85}{676}$

Step 3.5.1.3

Multiply $\frac{17}{26} \cdot \frac{85}{676}$ .

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Step 3.5.1.3.1

Multiply $\frac{17}{26}$ by $\frac{85}{676}$ .

$- \frac{1649}{17576} + \frac{289}{17576} + \frac{17 \cdot 85}{26 \cdot 676}$

Step 3.5.1.3.2

Multiply $17$ by $85$ .

$- \frac{1649}{17576} + \frac{289}{17576} + \frac{1445}{26 \cdot 676}$

Step 3.5.1.3.3

Multiply $26$ by $676$ .

$- \frac{1649}{17576} + \frac{289}{17576} + \frac{1445}{17576}$

Step 3.5.2

Combine the numerators over the common denominator.

$\frac{- 1649 + 289 + 1445}{17576}$

Step 3.5.3

Add $- 1649$ and $289$ .

$\frac{- 1360 + 1445}{17576}$

Step 3.5.4

Add $- 1360$ and $1445$ .

$\frac{85}{17576}$

Step 4

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

Step 5

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

$[\begin{array}{c} \frac{17}{26} & \frac{17}{26} & \frac{17}{26} & 1 & 0 & 0 \\ \frac{17}{26} & \frac{6}{13} & \frac{17}{26} & 0 & 1 & 0 \\ \frac{17}{26} & \frac{17}{26} & \frac{8}{13} & 0 & 0 & 1 \end{array}]$

Step 6

Find the reduced row echelon form.

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Step 6.1

Multiply each element of $R_{1}$ by $\frac{26}{17}$ to make the entry at $1, 1$ a $1$ .

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Step 6.1.1

Multiply each element of $R_{1}$ by $\frac{26}{17}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{26}{17} \cdot \frac{17}{26} & \frac{26}{17} \cdot \frac{17}{26} & \frac{26}{17} \cdot \frac{17}{26} & \frac{26}{17} \cdot 1 & \frac{26}{17} \cdot 0 & \frac{26}{17} \cdot 0 \\ \frac{17}{26} & \frac{6}{13} & \frac{17}{26} & 0 & 1 & 0 \\ \frac{17}{26} & \frac{17}{26} & \frac{8}{13} & 0 & 0 & 1 \end{array}]$

Step 6.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 1 & \frac{26}{17} & 0 & 0 \\ \frac{17}{26} & \frac{6}{13} & \frac{17}{26} & 0 & 1 & 0 \\ \frac{17}{26} & \frac{17}{26} & \frac{8}{13} & 0 & 0 & 1 \end{array}]$

Step 6.2

Perform the row operation $R_{2} = R_{2} - \frac{17}{26} R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 6.2.1

Perform the row operation $R_{2} = R_{2} - \frac{17}{26} R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & \frac{26}{17} & 0 & 0 \\ \frac{17}{26} - \frac{17}{26} \cdot 1 & \frac{6}{13} - \frac{17}{26} \cdot 1 & \frac{17}{26} - \frac{17}{26} \cdot 1 & 0 - \frac{17}{26} \cdot \frac{26}{17} & 1 - \frac{17}{26} \cdot 0 & 0 - \frac{17}{26} \cdot 0 \\ \frac{17}{26} & \frac{17}{26} & \frac{8}{13} & 0 & 0 & 1 \end{array}]$

Step 6.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & \frac{26}{17} & 0 & 0 \\ 0 & - \frac{5}{26} & 0 & - 1 & 1 & 0 \\ \frac{17}{26} & \frac{17}{26} & \frac{8}{13} & 0 & 0 & 1 \end{array}]$

Step 6.3

Perform the row operation $R_{3} = R_{3} - \frac{17}{26} R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 6.3.1

Perform the row operation $R_{3} = R_{3} - \frac{17}{26} R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & \frac{26}{17} & 0 & 0 \\ 0 & - \frac{5}{26} & 0 & - 1 & 1 & 0 \\ \frac{17}{26} - \frac{17}{26} \cdot 1 & \frac{17}{26} - \frac{17}{26} \cdot 1 & \frac{8}{13} - \frac{17}{26} \cdot 1 & 0 - \frac{17}{26} \cdot \frac{26}{17} & 0 - \frac{17}{26} \cdot 0 & 1 - \frac{17}{26} \cdot 0 \end{array}]$

Step 6.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & \frac{26}{17} & 0 & 0 \\ 0 & - \frac{5}{26} & 0 & - 1 & 1 & 0 \\ 0 & 0 & - \frac{1}{26} & - 1 & 0 & 1 \end{array}]$

Step 6.4

Multiply each element of $R_{2}$ by $- \frac{26}{5}$ to make the entry at $2, 2$ a $1$ .

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Step 6.4.1

Multiply each element of $R_{2}$ by $- \frac{26}{5}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & \frac{26}{17} & 0 & 0 \\ - \frac{26}{5} \cdot 0 & - \frac{26}{5} (- \frac{5}{26}) & - \frac{26}{5} \cdot 0 & - \frac{26}{5} \cdot - 1 & - \frac{26}{5} \cdot 1 & - \frac{26}{5} \cdot 0 \\ 0 & 0 & - \frac{1}{26} & - 1 & 0 & 1 \end{array}]$

Step 6.4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & \frac{26}{17} & 0 & 0 \\ 0 & 1 & 0 & \frac{26}{5} & - \frac{26}{5} & 0 \\ 0 & 0 & - \frac{1}{26} & - 1 & 0 & 1 \end{array}]$

Step 6.5

Multiply each element of $R_{3}$ by $- 26$ to make the entry at $3, 3$ a $1$ .

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Step 6.5.1

Multiply each element of $R_{3}$ by $- 26$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & \frac{26}{17} & 0 & 0 \\ 0 & 1 & 0 & \frac{26}{5} & - \frac{26}{5} & 0 \\ - 26 \cdot 0 & - 26 \cdot 0 & - 26 (- \frac{1}{26}) & - 26 \cdot - 1 & - 26 \cdot 0 & - 26 \cdot 1 \end{array}]$

Step 6.5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & \frac{26}{17} & 0 & 0 \\ 0 & 1 & 0 & \frac{26}{5} & - \frac{26}{5} & 0 \\ 0 & 0 & 1 & 26 & 0 & - 26 \end{array}]$

Step 6.6

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

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Step 6.6.1

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

$[\begin{array}{c} 1 - 0 & 1 - 0 & 1 - 1 & \frac{26}{17} - 26 & 0 - 0 & 0 + 26 \\ 0 & 1 & 0 & \frac{26}{5} & - \frac{26}{5} & 0 \\ 0 & 0 & 1 & 26 & 0 & - 26 \end{array}]$

Step 6.6.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & - \frac{416}{17} & 0 & 26 \\ 0 & 1 & 0 & \frac{26}{5} & - \frac{26}{5} & 0 \\ 0 & 0 & 1 & 26 & 0 & - 26 \end{array}]$

Step 6.7

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

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Step 6.7.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & - \frac{416}{17} - \frac{26}{5} & 0 + \frac{26}{5} & 26 - 0 \\ 0 & 1 & 0 & \frac{26}{5} & - \frac{26}{5} & 0 \\ 0 & 0 & 1 & 26 & 0 & - 26 \end{array}]$

Step 6.7.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - \frac{2522}{85} & \frac{26}{5} & 26 \\ 0 & 1 & 0 & \frac{26}{5} & - \frac{26}{5} & 0 \\ 0 & 0 & 1 & 26 & 0 & - 26 \end{array}]$

Step 7

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

$[\begin{array}{c} - \frac{2522}{85} & \frac{26}{5} & 26 \\ \frac{26}{5} & - \frac{26}{5} & 0 \\ 26 & 0 & - 26 \end{array}]$