Finite Math Examples

$[\begin{array}{c} \frac{2}{3} & \frac{1}{4} \\ \frac{1}{3} & \frac{3}{4} \end{array}]$

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

$\frac{2}{3} \cdot \frac{3}{4} - \frac{1}{3} \cdot \frac{1}{4}$

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 $4$ .

$\frac{2}{3} \cdot \frac{3}{2 (2)} - \frac{1}{3} \cdot \frac{1}{4}$

Step 2.2.1.1.2

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3}{2 \cdot 2} - \frac{1}{3} \cdot \frac{1}{4}$

Step 2.2.1.1.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{3}{2} - \frac{1}{3} \cdot \frac{1}{4}$

Step 2.2.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.2.1

Cancel the common factor.

$\frac{1}{3} \cdot \frac{3}{2} - \frac{1}{3} \cdot \frac{1}{4}$

Step 2.2.1.2.2

Rewrite the expression.

$\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{4}$

Step 2.2.1.3

Multiply $- \frac{1}{3} \cdot \frac{1}{4}$ .

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Step 2.2.1.3.1

Multiply $\frac{1}{4}$ by $\frac{1}{3}$ .

$\frac{1}{2} - \frac{1}{4 \cdot 3}$

Step 2.2.1.3.2

Multiply $4$ by $3$ .

$\frac{1}{2} - \frac{1}{12}$

Step 2.2.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{1}{2} \cdot \frac{6}{6} - \frac{1}{12}$

Step 2.2.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.3.1

Multiply $\frac{1}{2}$ by $\frac{6}{6}$ .

$\frac{6}{2 \cdot 6} - \frac{1}{12}$

Step 2.2.3.2

Multiply $2$ by $6$ .

$\frac{6}{12} - \frac{1}{12}$

Step 2.2.4

Combine the numerators over the common denominator.

$\frac{6 - 1}{12}$

Step 2.2.5

Subtract $1$ from $6$ .

$\frac{5}{12}$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{\frac{5}{12}} [\begin{array}{c} \frac{3}{4} & - \frac{1}{4} \\ - \frac{1}{3} & \frac{2}{3} \end{array}]$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{12}{5}) [\begin{array}{c} \frac{3}{4} & - \frac{1}{4} \\ - \frac{1}{3} & \frac{2}{3} \end{array}]$

Step 6

Multiply $\frac{12}{5}$ by $1$ .

$\frac{12}{5} [\begin{array}{c} \frac{3}{4} & - \frac{1}{4} \\ - \frac{1}{3} & \frac{2}{3} \end{array}]$

Step 7

Multiply $\frac{12}{5}$ by each element of the matrix.

$[\begin{array}{c} \frac{12}{5} \cdot \frac{3}{4} & \frac{12}{5} (- \frac{1}{4}) \\ \frac{12}{5} (- \frac{1}{3}) & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8

Simplify each element in the matrix.

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Step 8.1

Cancel the common factor of $4$ .

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Step 8.1.1

Factor $4$ out of $12$ .

$[\begin{array}{c} \frac{4 (3)}{5} \cdot \frac{3}{4} & \frac{12}{5} (- \frac{1}{4}) \\ \frac{12}{5} (- \frac{1}{3}) & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.1.2

Cancel the common factor.

$[\begin{array}{c} \frac{4 \cdot 3}{5} \cdot \frac{3}{4} & \frac{12}{5} (- \frac{1}{4}) \\ \frac{12}{5} (- \frac{1}{3}) & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.1.3

Rewrite the expression.

$[\begin{array}{c} \frac{3}{5} \cdot 3 & \frac{12}{5} (- \frac{1}{4}) \\ \frac{12}{5} (- \frac{1}{3}) & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.2

Combine $\frac{3}{5}$ and $3$ .

$[\begin{array}{c} \frac{3 \cdot 3}{5} & \frac{12}{5} (- \frac{1}{4}) \\ \frac{12}{5} (- \frac{1}{3}) & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.3

Multiply $3$ by $3$ .

$[\begin{array}{c} \frac{9}{5} & \frac{12}{5} (- \frac{1}{4}) \\ \frac{12}{5} (- \frac{1}{3}) & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.4

Cancel the common factor of $4$ .

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Step 8.4.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$[\begin{array}{c} \frac{9}{5} & \frac{12}{5} \cdot \frac{- 1}{4} \\ \frac{12}{5} (- \frac{1}{3}) & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.4.2

Factor $4$ out of $12$ .

$[\begin{array}{c} \frac{9}{5} & \frac{4 (3)}{5} \cdot \frac{- 1}{4} \\ \frac{12}{5} (- \frac{1}{3}) & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.4.3

Cancel the common factor.

$[\begin{array}{c} \frac{9}{5} & \frac{4 \cdot 3}{5} \cdot \frac{- 1}{4} \\ \frac{12}{5} (- \frac{1}{3}) & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.4.4

Rewrite the expression.

$[\begin{array}{c} \frac{9}{5} & \frac{3}{5} \cdot - 1 \\ \frac{12}{5} (- \frac{1}{3}) & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.5

Combine $\frac{3}{5}$ and $- 1$ .

$[\begin{array}{c} \frac{9}{5} & \frac{3 \cdot - 1}{5} \\ \frac{12}{5} (- \frac{1}{3}) & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.6

Multiply $3$ by $- 1$ .

$[\begin{array}{c} \frac{9}{5} & \frac{- 3}{5} \\ \frac{12}{5} (- \frac{1}{3}) & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.7

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{9}{5} & - \frac{3}{5} \\ \frac{12}{5} (- \frac{1}{3}) & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.8

Cancel the common factor of $3$ .

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Step 8.8.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$[\begin{array}{c} \frac{9}{5} & - \frac{3}{5} \\ \frac{12}{5} \cdot \frac{- 1}{3} & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.8.2

Factor $3$ out of $12$ .

$[\begin{array}{c} \frac{9}{5} & - \frac{3}{5} \\ \frac{3 (4)}{5} \cdot \frac{- 1}{3} & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.8.3

Cancel the common factor.

$[\begin{array}{c} \frac{9}{5} & - \frac{3}{5} \\ \frac{3 \cdot 4}{5} \cdot \frac{- 1}{3} & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.8.4

Rewrite the expression.

$[\begin{array}{c} \frac{9}{5} & - \frac{3}{5} \\ \frac{4}{5} \cdot - 1 & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.9

Combine $\frac{4}{5}$ and $- 1$ .

$[\begin{array}{c} \frac{9}{5} & - \frac{3}{5} \\ \frac{4 \cdot - 1}{5} & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.10

Multiply $4$ by $- 1$ .

$[\begin{array}{c} \frac{9}{5} & - \frac{3}{5} \\ \frac{- 4}{5} & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.11

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{9}{5} & - \frac{3}{5} \\ - \frac{4}{5} & \frac{12}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.12

Cancel the common factor of $3$ .

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Step 8.12.1

Factor $3$ out of $12$ .

$[\begin{array}{c} \frac{9}{5} & - \frac{3}{5} \\ - \frac{4}{5} & \frac{3 (4)}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.12.2

Cancel the common factor.

$[\begin{array}{c} \frac{9}{5} & - \frac{3}{5} \\ - \frac{4}{5} & \frac{3 \cdot 4}{5} \cdot \frac{2}{3} \end{array}]$

Step 8.12.3

Rewrite the expression.

$[\begin{array}{c} \frac{9}{5} & - \frac{3}{5} \\ - \frac{4}{5} & \frac{4}{5} \cdot 2 \end{array}]$

Step 8.13

Combine $\frac{4}{5}$ and $2$ .

$[\begin{array}{c} \frac{9}{5} & - \frac{3}{5} \\ - \frac{4}{5} & \frac{4 \cdot 2}{5} \end{array}]$

Step 8.14

Multiply $4$ by $2$ .

$[\begin{array}{c} \frac{9}{5} & - \frac{3}{5} \\ - \frac{4}{5} & \frac{8}{5} \end{array}]$