Algebra Examples

$[\begin{array}{c} \frac{40}{13} & \frac{70}{13} \\ \frac{28}{13} & \frac{140}{13} \end{array}]$

Step 1

The inverse of a $2 \times 2$ matrix can be found using the formula $\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where $a d - b c$ is the determinant.

Step 2

Find the determinant.

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Step 2.1

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

$\frac{40}{13} \cdot \frac{140}{13} - \frac{28}{13} \cdot \frac{70}{13}$

Step 2.2

Simplify the determinant.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $\frac{40}{13} \cdot \frac{140}{13}$ .

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Step 2.2.1.1.1

Multiply $\frac{40}{13}$ by $\frac{140}{13}$ .

$\frac{40 \cdot 140}{13 \cdot 13} - \frac{28}{13} \cdot \frac{70}{13}$

Step 2.2.1.1.2

Multiply $40$ by $140$ .

$\frac{5600}{13 \cdot 13} - \frac{28}{13} \cdot \frac{70}{13}$

Step 2.2.1.1.3

Multiply $13$ by $13$ .

$\frac{5600}{169} - \frac{28}{13} \cdot \frac{70}{13}$

Step 2.2.1.2

Multiply $- \frac{28}{13} \cdot \frac{70}{13}$ .

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Step 2.2.1.2.1

Multiply $\frac{70}{13}$ by $\frac{28}{13}$ .

$\frac{5600}{169} - \frac{70 \cdot 28}{13 \cdot 13}$

Step 2.2.1.2.2

Multiply $70$ by $28$ .

$\frac{5600}{169} - \frac{1960}{13 \cdot 13}$

Step 2.2.1.2.3

Multiply $13$ by $13$ .

$\frac{5600}{169} - \frac{1960}{169}$

Step 2.2.2

Combine the numerators over the common denominator.

$\frac{5600 - 1960}{169}$

Step 2.2.3

Subtract $1960$ from $5600$ .

$\frac{3640}{169}$

Step 2.2.4

Cancel the common factor of $3640$ and $169$ .

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Step 2.2.4.1

Factor $13$ out of $3640$ .

$\frac{13 (280)}{169}$

Step 2.2.4.2

Cancel the common factors.

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Step 2.2.4.2.1

Factor $13$ out of $169$ .

$\frac{13 \cdot 280}{13 \cdot 13}$

Step 2.2.4.2.2

Cancel the common factor.

$\frac{13 \cdot 280}{13 \cdot 13}$

Step 2.2.4.2.3

Rewrite the expression.

$\frac{280}{13}$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{\frac{280}{13}} [\begin{array}{c} \frac{140}{13} & - \frac{70}{13} \\ - \frac{28}{13} & \frac{40}{13} \end{array}]$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{13}{280}) [\begin{array}{c} \frac{140}{13} & - \frac{70}{13} \\ - \frac{28}{13} & \frac{40}{13} \end{array}]$

Step 6

Multiply $\frac{13}{280}$ by $1$ .

$\frac{13}{280} [\begin{array}{c} \frac{140}{13} & - \frac{70}{13} \\ - \frac{28}{13} & \frac{40}{13} \end{array}]$

Step 7

Multiply $\frac{13}{280}$ by each element of the matrix.

$[\begin{array}{c} \frac{13}{280} \cdot \frac{140}{13} & \frac{13}{280} (- \frac{70}{13}) \\ \frac{13}{280} (- \frac{28}{13}) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8

Simplify each element in the matrix.

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Step 8.1

Cancel the common factor of $13$ .

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Step 8.1.1

Cancel the common factor.

$[\begin{array}{c} \frac{13}{280} \cdot \frac{140}{13} & \frac{13}{280} (- \frac{70}{13}) \\ \frac{13}{280} (- \frac{28}{13}) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.1.2

Rewrite the expression.

$[\begin{array}{c} \frac{1}{280} \cdot 140 & \frac{13}{280} (- \frac{70}{13}) \\ \frac{13}{280} (- \frac{28}{13}) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.2

Cancel the common factor of $140$ .

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Step 8.2.1

Factor $140$ out of $280$ .

$[\begin{array}{c} \frac{1}{140 (2)} \cdot 140 & \frac{13}{280} (- \frac{70}{13}) \\ \frac{13}{280} (- \frac{28}{13}) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.2.2

Cancel the common factor.

$[\begin{array}{c} \frac{1}{140 \cdot 2} \cdot 140 & \frac{13}{280} (- \frac{70}{13}) \\ \frac{13}{280} (- \frac{28}{13}) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.2.3

Rewrite the expression.

$[\begin{array}{c} \frac{1}{2} & \frac{13}{280} (- \frac{70}{13}) \\ \frac{13}{280} (- \frac{28}{13}) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.3

Cancel the common factor of $13$ .

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Step 8.3.1

Move the leading negative in $- \frac{70}{13}$ into the numerator.

$[\begin{array}{c} \frac{1}{2} & \frac{13}{280} \cdot \frac{- 70}{13} \\ \frac{13}{280} (- \frac{28}{13}) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.3.2

Cancel the common factor.

$[\begin{array}{c} \frac{1}{2} & \frac{13}{280} \cdot \frac{- 70}{13} \\ \frac{13}{280} (- \frac{28}{13}) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.3.3

Rewrite the expression.

$[\begin{array}{c} \frac{1}{2} & \frac{1}{280} \cdot - 70 \\ \frac{13}{280} (- \frac{28}{13}) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.4

Cancel the common factor of $70$ .

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Step 8.4.1

Factor $70$ out of $280$ .

$[\begin{array}{c} \frac{1}{2} & \frac{1}{70 (4)} \cdot - 70 \\ \frac{13}{280} (- \frac{28}{13}) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.4.2

Factor $70$ out of $- 70$ .

$[\begin{array}{c} \frac{1}{2} & \frac{1}{70 \cdot 4} \cdot (70 \cdot - 1) \\ \frac{13}{280} (- \frac{28}{13}) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.4.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{2} & \frac{1}{70 \cdot 4} \cdot (70 \cdot - 1) \\ \frac{13}{280} (- \frac{28}{13}) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.4.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{2} & \frac{1}{4} \cdot - 1 \\ \frac{13}{280} (- \frac{28}{13}) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.5

Combine $\frac{1}{4}$ and $- 1$ .

$[\begin{array}{c} \frac{1}{2} & \frac{- 1}{4} \\ \frac{13}{280} (- \frac{28}{13}) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.6

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{4} \\ \frac{13}{280} (- \frac{28}{13}) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.7

Cancel the common factor of $13$ .

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Step 8.7.1

Move the leading negative in $- \frac{28}{13}$ into the numerator.

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{4} \\ \frac{13}{280} \cdot \frac{- 28}{13} & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.7.2

Cancel the common factor.

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{4} \\ \frac{13}{280} \cdot \frac{- 28}{13} & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.7.3

Rewrite the expression.

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{4} \\ \frac{1}{280} \cdot - 28 & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.8

Cancel the common factor of $28$ .

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Step 8.8.1

Factor $28$ out of $280$ .

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{4} \\ \frac{1}{28 (10)} \cdot - 28 & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.8.2

Factor $28$ out of $- 28$ .

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{4} \\ \frac{1}{28 \cdot 10} \cdot (28 \cdot - 1) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.8.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{4} \\ \frac{1}{28 \cdot 10} \cdot (28 \cdot - 1) & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.8.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{4} \\ \frac{1}{10} \cdot - 1 & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.9

Combine $\frac{1}{10}$ and $- 1$ .

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{4} \\ \frac{- 1}{10} & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.10

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{4} \\ - \frac{1}{10} & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.11

Cancel the common factor of $13$ .

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Step 8.11.1

Cancel the common factor.

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{4} \\ - \frac{1}{10} & \frac{13}{280} \cdot \frac{40}{13} \end{array}]$

Step 8.11.2

Rewrite the expression.

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{4} \\ - \frac{1}{10} & \frac{1}{280} \cdot 40 \end{array}]$

Step 8.12

Cancel the common factor of $40$ .

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Step 8.12.1

Factor $40$ out of $280$ .

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{4} \\ - \frac{1}{10} & \frac{1}{40 (7)} \cdot 40 \end{array}]$

Step 8.12.2

Cancel the common factor.

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{4} \\ - \frac{1}{10} & \frac{1}{40 \cdot 7} \cdot 40 \end{array}]$

Step 8.12.3

Rewrite the expression.

$[\begin{array}{c} \frac{1}{2} & - \frac{1}{4} \\ - \frac{1}{10} & \frac{1}{7} \end{array}]$