Finite Math Examples

$3 X = [\begin{array}{c} \frac{17}{22} & \frac{31}{110} \\ - \frac{4}{11} & \frac{16}{55} \end{array}]$

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

$\frac{17}{22} \cdot \frac{16}{55} - (- \frac{4}{11} \cdot \frac{31}{110})$

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

Cancel the common factor of $2$ .

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Step 2.2.1.1.1

Factor $2$ out of $22$ .

$\frac{17}{2 (11)} \cdot \frac{16}{55} - (- \frac{4}{11} \cdot \frac{31}{110})$

Step 2.2.1.1.2

Factor $2$ out of $16$ .

$\frac{17}{2 \cdot 11} \cdot \frac{2 \cdot 8}{55} - (- \frac{4}{11} \cdot \frac{31}{110})$

Step 2.2.1.1.3

Cancel the common factor.

$\frac{17}{2 \cdot 11} \cdot \frac{2 \cdot 8}{55} - (- \frac{4}{11} \cdot \frac{31}{110})$

Step 2.2.1.1.4

Rewrite the expression.

$\frac{17}{11} \cdot \frac{8}{55} - (- \frac{4}{11} \cdot \frac{31}{110})$

Step 2.2.1.2

Multiply $\frac{17}{11}$ by $\frac{8}{55}$ .

$\frac{17 \cdot 8}{11 \cdot 55} - (- \frac{4}{11} \cdot \frac{31}{110})$

Step 2.2.1.3

Multiply $17$ by $8$ .

$\frac{136}{11 \cdot 55} - (- \frac{4}{11} \cdot \frac{31}{110})$

Step 2.2.1.4

Multiply $11$ by $55$ .

$\frac{136}{605} - (- \frac{4}{11} \cdot \frac{31}{110})$

Step 2.2.1.5

Cancel the common factor of $2$ .

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Step 2.2.1.5.1

Move the leading negative in $- \frac{4}{11}$ into the numerator.

$\frac{136}{605} - (\frac{- 4}{11} \cdot \frac{31}{110})$

Step 2.2.1.5.2

Factor $2$ out of $- 4$ .

$\frac{136}{605} - (\frac{2 (- 2)}{11} \cdot \frac{31}{110})$

Step 2.2.1.5.3

Factor $2$ out of $110$ .

$\frac{136}{605} - (\frac{2 \cdot - 2}{11} \cdot \frac{31}{2 \cdot 55})$

Step 2.2.1.5.4

Cancel the common factor.

$\frac{136}{605} - (\frac{2 \cdot - 2}{11} \cdot \frac{31}{2 \cdot 55})$

Step 2.2.1.5.5

Rewrite the expression.

$\frac{136}{605} - (\frac{- 2}{11} \cdot \frac{31}{55})$

Step 2.2.1.6

Multiply $\frac{- 2}{11}$ by $\frac{31}{55}$ .

$\frac{136}{605} - \frac{- 2 \cdot 31}{11 \cdot 55}$

Step 2.2.1.7

Multiply $- 2$ by $31$ .

$\frac{136}{605} - \frac{- 62}{11 \cdot 55}$

Step 2.2.1.8

Multiply $11$ by $55$ .

$\frac{136}{605} - \frac{- 62}{605}$

Step 2.2.1.9

Move the negative in front of the fraction.

$\frac{136}{605} - - \frac{62}{605}$

Step 2.2.1.10

Multiply $- - \frac{62}{605}$ .

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Step 2.2.1.10.1

Multiply $- 1$ by $- 1$ .

$\frac{136}{605} + 1 (\frac{62}{605})$

Step 2.2.1.10.2

Multiply $\frac{62}{605}$ by $1$ .

$\frac{136}{605} + \frac{62}{605}$

Step 2.2.2

Combine the numerators over the common denominator.

$\frac{136 + 62}{605}$

Step 2.2.3

Add $136$ and $62$ .

$\frac{198}{605}$

Step 2.2.4

Cancel the common factor of $198$ and $605$ .

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Step 2.2.4.1

Factor $11$ out of $198$ .

$\frac{11 (18)}{605}$

Step 2.2.4.2

Cancel the common factors.

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Step 2.2.4.2.1

Factor $11$ out of $605$ .

$\frac{11 \cdot 18}{11 \cdot 55}$

Step 2.2.4.2.2

Cancel the common factor.

$\frac{11 \cdot 18}{11 \cdot 55}$

Step 2.2.4.2.3

Rewrite the expression.

$\frac{18}{55}$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{\frac{18}{55}} [\begin{array}{c} \frac{16}{55} & - \frac{31}{110} \\ \frac{4}{11} & \frac{17}{22} \end{array}]$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{55}{18}) [\begin{array}{c} \frac{16}{55} & - \frac{31}{110} \\ \frac{4}{11} & \frac{17}{22} \end{array}]$

Step 6

Multiply $\frac{55}{18}$ by $1$ .

$\frac{55}{18} [\begin{array}{c} \frac{16}{55} & - \frac{31}{110} \\ \frac{4}{11} & \frac{17}{22} \end{array}]$

Step 7

Multiply $\frac{55}{18}$ by each element of the matrix.

$[\begin{array}{c} \frac{55}{18} \cdot \frac{16}{55} & \frac{55}{18} (- \frac{31}{110}) \\ \frac{55}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8

Simplify each element in the matrix.

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Step 8.1

Cancel the common factor of $55$ .

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Step 8.1.1

Cancel the common factor.

$[\begin{array}{c} \frac{55}{18} \cdot \frac{16}{55} & \frac{55}{18} (- \frac{31}{110}) \\ \frac{55}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.1.2

Rewrite the expression.

$[\begin{array}{c} \frac{1}{18} \cdot 16 & \frac{55}{18} (- \frac{31}{110}) \\ \frac{55}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.2

Cancel the common factor of $2$ .

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Step 8.2.1

Factor $2$ out of $18$ .

$[\begin{array}{c} \frac{1}{2 (9)} \cdot 16 & \frac{55}{18} (- \frac{31}{110}) \\ \frac{55}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.2.2

Factor $2$ out of $16$ .

$[\begin{array}{c} \frac{1}{2 \cdot 9} \cdot (2 \cdot 8) & \frac{55}{18} (- \frac{31}{110}) \\ \frac{55}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.2.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{2 \cdot 9} \cdot (2 \cdot 8) & \frac{55}{18} (- \frac{31}{110}) \\ \frac{55}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.2.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{9} \cdot 8 & \frac{55}{18} (- \frac{31}{110}) \\ \frac{55}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.3

Combine $\frac{1}{9}$ and $8$ .

$[\begin{array}{c} \frac{8}{9} & \frac{55}{18} (- \frac{31}{110}) \\ \frac{55}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.4

Cancel the common factor of $55$ .

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Step 8.4.1

Move the leading negative in $- \frac{31}{110}$ into the numerator.

$[\begin{array}{c} \frac{8}{9} & \frac{55}{18} \cdot \frac{- 31}{110} \\ \frac{55}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.4.2

Factor $55$ out of $110$ .

$[\begin{array}{c} \frac{8}{9} & \frac{55}{18} \cdot \frac{- 31}{55 (2)} \\ \frac{55}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.4.3

Cancel the common factor.

$[\begin{array}{c} \frac{8}{9} & \frac{55}{18} \cdot \frac{- 31}{55 \cdot 2} \\ \frac{55}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.4.4

Rewrite the expression.

$[\begin{array}{c} \frac{8}{9} & \frac{1}{18} \cdot \frac{- 31}{2} \\ \frac{55}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.5

Multiply $\frac{1}{18}$ by $\frac{- 31}{2}$ .

$[\begin{array}{c} \frac{8}{9} & \frac{- 31}{18 \cdot 2} \\ \frac{55}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.6

Multiply $18$ by $2$ .

$[\begin{array}{c} \frac{8}{9} & \frac{- 31}{36} \\ \frac{55}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.7

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{55}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.8

Cancel the common factor of $11$ .

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Step 8.8.1

Factor $11$ out of $55$ .

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{11 (5)}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.8.2

Cancel the common factor.

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{11 \cdot 5}{18} \cdot \frac{4}{11} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.8.3

Rewrite the expression.

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{5}{18} \cdot 4 & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.9

Cancel the common factor of $2$ .

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Step 8.9.1

Factor $2$ out of $18$ .

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{5}{2 (9)} \cdot 4 & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.9.2

Factor $2$ out of $4$ .

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{5}{2 \cdot 9} \cdot (2 \cdot 2) & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.9.3

Cancel the common factor.

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{5}{2 \cdot 9} \cdot (2 \cdot 2) & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.9.4

Rewrite the expression.

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{5}{9} \cdot 2 & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.10

Combine $\frac{5}{9}$ and $2$ .

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{5 \cdot 2}{9} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.11

Multiply $5$ by $2$ .

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{10}{9} & \frac{55}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.12

Cancel the common factor of $11$ .

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Step 8.12.1

Factor $11$ out of $55$ .

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{10}{9} & \frac{11 (5)}{18} \cdot \frac{17}{22} \end{array}]$

Step 8.12.2

Factor $11$ out of $22$ .

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{10}{9} & \frac{11 \cdot 5}{18} \cdot \frac{17}{11 \cdot 2} \end{array}]$

Step 8.12.3

Cancel the common factor.

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{10}{9} & \frac{11 \cdot 5}{18} \cdot \frac{17}{11 \cdot 2} \end{array}]$

Step 8.12.4

Rewrite the expression.

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{10}{9} & \frac{5}{18} \cdot \frac{17}{2} \end{array}]$

Step 8.13

Multiply $\frac{5}{18}$ by $\frac{17}{2}$ .

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{10}{9} & \frac{5 \cdot 17}{18 \cdot 2} \end{array}]$

Step 8.14

Multiply $5$ by $17$ .

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{10}{9} & \frac{85}{18 \cdot 2} \end{array}]$

Step 8.15

Multiply $18$ by $2$ .

$[\begin{array}{c} \frac{8}{9} & - \frac{31}{36} \\ \frac{10}{9} & \frac{85}{36} \end{array}]$