Finite Math Examples

$[\begin{array}{c} - 3 & 3 \\ 8 & 7 \end{array}]$

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

$- 3 \cdot 7 - 8 \cdot 3$

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 $- 3$ by $7$ .

$- 21 - 8 \cdot 3$

Step 2.2.1.2

Multiply $- 8$ by $3$ .

$- 21 - 24$

Step 2.2.2

Subtract $24$ from $- 21$ .

$- 45$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- 45} [\begin{array}{c} 7 & - 3 \\ - 8 & - 3 \end{array}]$

Step 5

Move the negative in front of the fraction.

$- \frac{1}{45} [\begin{array}{c} 7 & - 3 \\ - 8 & - 3 \end{array}]$

Step 6

Multiply $- \frac{1}{45}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{45} \cdot 7 & - \frac{1}{45} \cdot - 3 \\ - \frac{1}{45} \cdot - 8 & - \frac{1}{45} \cdot - 3 \end{array}]$

Step 7

Simplify each element in the matrix.

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Step 7.1

Multiply $- \frac{1}{45} \cdot 7$ .

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Step 7.1.1

Multiply $7$ by $- 1$ .

$[\begin{array}{c} - 7 (\frac{1}{45}) & - \frac{1}{45} \cdot - 3 \\ - \frac{1}{45} \cdot - 8 & - \frac{1}{45} \cdot - 3 \end{array}]$

Step 7.1.2

Combine $- 7$ and $\frac{1}{45}$ .

$[\begin{array}{c} \frac{- 7}{45} & - \frac{1}{45} \cdot - 3 \\ - \frac{1}{45} \cdot - 8 & - \frac{1}{45} \cdot - 3 \end{array}]$

Step 7.2

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{7}{45} & - \frac{1}{45} \cdot - 3 \\ - \frac{1}{45} \cdot - 8 & - \frac{1}{45} \cdot - 3 \end{array}]$

Step 7.3

Cancel the common factor of $3$ .

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Step 7.3.1

Move the leading negative in $- \frac{1}{45}$ into the numerator.

$[\begin{array}{c} - \frac{7}{45} & \frac{- 1}{45} \cdot - 3 \\ - \frac{1}{45} \cdot - 8 & - \frac{1}{45} \cdot - 3 \end{array}]$

Step 7.3.2

Factor $3$ out of $45$ .

$[\begin{array}{c} - \frac{7}{45} & \frac{- 1}{3 (15)} \cdot - 3 \\ - \frac{1}{45} \cdot - 8 & - \frac{1}{45} \cdot - 3 \end{array}]$

Step 7.3.3

Factor $3$ out of $- 3$ .

$[\begin{array}{c} - \frac{7}{45} & \frac{- 1}{3 \cdot 15} \cdot (3 \cdot - 1) \\ - \frac{1}{45} \cdot - 8 & - \frac{1}{45} \cdot - 3 \end{array}]$

Step 7.3.4

Cancel the common factor.

$[\begin{array}{c} - \frac{7}{45} & \frac{- 1}{3 \cdot 15} \cdot (3 \cdot - 1) \\ - \frac{1}{45} \cdot - 8 & - \frac{1}{45} \cdot - 3 \end{array}]$

Step 7.3.5

Rewrite the expression.

$[\begin{array}{c} - \frac{7}{45} & \frac{- 1}{15} \cdot - 1 \\ - \frac{1}{45} \cdot - 8 & - \frac{1}{45} \cdot - 3 \end{array}]$

Step 7.4

Combine $\frac{- 1}{15}$ and $- 1$ .

$[\begin{array}{c} - \frac{7}{45} & \frac{- - 1}{15} \\ - \frac{1}{45} \cdot - 8 & - \frac{1}{45} \cdot - 3 \end{array}]$

Step 7.5

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} - \frac{7}{45} & \frac{1}{15} \\ - \frac{1}{45} \cdot - 8 & - \frac{1}{45} \cdot - 3 \end{array}]$

Step 7.6

Multiply $- \frac{1}{45} \cdot - 8$ .

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Step 7.6.1

Multiply $- 8$ by $- 1$ .

$[\begin{array}{c} - \frac{7}{45} & \frac{1}{15} \\ 8 (\frac{1}{45}) & - \frac{1}{45} \cdot - 3 \end{array}]$

Step 7.6.2

Combine $8$ and $\frac{1}{45}$ .

$[\begin{array}{c} - \frac{7}{45} & \frac{1}{15} \\ \frac{8}{45} & - \frac{1}{45} \cdot - 3 \end{array}]$

Step 7.7

Cancel the common factor of $3$ .

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Step 7.7.1

Move the leading negative in $- \frac{1}{45}$ into the numerator.

$[\begin{array}{c} - \frac{7}{45} & \frac{1}{15} \\ \frac{8}{45} & \frac{- 1}{45} \cdot - 3 \end{array}]$

Step 7.7.2

Factor $3$ out of $45$ .

$[\begin{array}{c} - \frac{7}{45} & \frac{1}{15} \\ \frac{8}{45} & \frac{- 1}{3 (15)} \cdot - 3 \end{array}]$

Step 7.7.3

Factor $3$ out of $- 3$ .

$[\begin{array}{c} - \frac{7}{45} & \frac{1}{15} \\ \frac{8}{45} & \frac{- 1}{3 \cdot 15} \cdot (3 \cdot - 1) \end{array}]$

Step 7.7.4

Cancel the common factor.

$[\begin{array}{c} - \frac{7}{45} & \frac{1}{15} \\ \frac{8}{45} & \frac{- 1}{3 \cdot 15} \cdot (3 \cdot - 1) \end{array}]$

Step 7.7.5

Rewrite the expression.

$[\begin{array}{c} - \frac{7}{45} & \frac{1}{15} \\ \frac{8}{45} & \frac{- 1}{15} \cdot - 1 \end{array}]$

Step 7.8

Combine $\frac{- 1}{15}$ and $- 1$ .

$[\begin{array}{c} - \frac{7}{45} & \frac{1}{15} \\ \frac{8}{45} & \frac{- - 1}{15} \end{array}]$

Step 7.9

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} - \frac{7}{45} & \frac{1}{15} \\ \frac{8}{45} & \frac{1}{15} \end{array}]$