Finite Math Examples

$[\begin{array}{c} 6 & 4 \\ 2 & 7 \end{array}]$

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

$6 \cdot 7 - 2 \cdot 4$

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

Multiply $6$ by $7$ .

$42 - 2 \cdot 4$

Step 2.2.1.2

Multiply $- 2$ by $4$ .

$42 - 8$

Step 2.2.2

Subtract $8$ from $42$ .

$34$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{34} [\begin{array}{c} 7 & - 4 \\ - 2 & 6 \end{array}]$

Step 5

Multiply $\frac{1}{34}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{34} \cdot 7 & \frac{1}{34} \cdot - 4 \\ \frac{1}{34} \cdot - 2 & \frac{1}{34} \cdot 6 \end{array}]$

Step 6

Simplify each element in the matrix.

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Step 6.1

Combine $\frac{1}{34}$ and $7$ .

$[\begin{array}{c} \frac{7}{34} & \frac{1}{34} \cdot - 4 \\ \frac{1}{34} \cdot - 2 & \frac{1}{34} \cdot 6 \end{array}]$

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Factor $2$ out of $34$ .

$[\begin{array}{c} \frac{7}{34} & \frac{1}{2 (17)} \cdot - 4 \\ \frac{1}{34} \cdot - 2 & \frac{1}{34} \cdot 6 \end{array}]$

Step 6.2.2

Factor $2$ out of $- 4$ .

$[\begin{array}{c} \frac{7}{34} & \frac{1}{2 \cdot 17} \cdot (2 \cdot - 2) \\ \frac{1}{34} \cdot - 2 & \frac{1}{34} \cdot 6 \end{array}]$

Step 6.2.3

Cancel the common factor.

$[\begin{array}{c} \frac{7}{34} & \frac{1}{2 \cdot 17} \cdot (2 \cdot - 2) \\ \frac{1}{34} \cdot - 2 & \frac{1}{34} \cdot 6 \end{array}]$

Step 6.2.4

Rewrite the expression.

$[\begin{array}{c} \frac{7}{34} & \frac{1}{17} \cdot - 2 \\ \frac{1}{34} \cdot - 2 & \frac{1}{34} \cdot 6 \end{array}]$

Step 6.3

Combine $\frac{1}{17}$ and $- 2$ .

$[\begin{array}{c} \frac{7}{34} & \frac{- 2}{17} \\ \frac{1}{34} \cdot - 2 & \frac{1}{34} \cdot 6 \end{array}]$

Step 6.4

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{7}{34} & - \frac{2}{17} \\ \frac{1}{34} \cdot - 2 & \frac{1}{34} \cdot 6 \end{array}]$

Step 6.5

Cancel the common factor of $2$ .

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Step 6.5.1

Factor $2$ out of $34$ .

$[\begin{array}{c} \frac{7}{34} & - \frac{2}{17} \\ \frac{1}{2 (17)} \cdot - 2 & \frac{1}{34} \cdot 6 \end{array}]$

Step 6.5.2

Factor $2$ out of $- 2$ .

$[\begin{array}{c} \frac{7}{34} & - \frac{2}{17} \\ \frac{1}{2 \cdot 17} \cdot (2 \cdot - 1) & \frac{1}{34} \cdot 6 \end{array}]$

Step 6.5.3

Cancel the common factor.

$[\begin{array}{c} \frac{7}{34} & - \frac{2}{17} \\ \frac{1}{2 \cdot 17} \cdot (2 \cdot - 1) & \frac{1}{34} \cdot 6 \end{array}]$

Step 6.5.4

Rewrite the expression.

$[\begin{array}{c} \frac{7}{34} & - \frac{2}{17} \\ \frac{1}{17} \cdot - 1 & \frac{1}{34} \cdot 6 \end{array}]$

Step 6.6

Combine $\frac{1}{17}$ and $- 1$ .

$[\begin{array}{c} \frac{7}{34} & - \frac{2}{17} \\ \frac{- 1}{17} & \frac{1}{34} \cdot 6 \end{array}]$

Step 6.7

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{7}{34} & - \frac{2}{17} \\ - \frac{1}{17} & \frac{1}{34} \cdot 6 \end{array}]$

Step 6.8

Cancel the common factor of $2$ .

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Step 6.8.1

Factor $2$ out of $34$ .

$[\begin{array}{c} \frac{7}{34} & - \frac{2}{17} \\ - \frac{1}{17} & \frac{1}{2 (17)} \cdot 6 \end{array}]$

Step 6.8.2

Factor $2$ out of $6$ .

$[\begin{array}{c} \frac{7}{34} & - \frac{2}{17} \\ - \frac{1}{17} & \frac{1}{2 \cdot 17} \cdot (2 \cdot 3) \end{array}]$

Step 6.8.3

Cancel the common factor.

$[\begin{array}{c} \frac{7}{34} & - \frac{2}{17} \\ - \frac{1}{17} & \frac{1}{2 \cdot 17} \cdot (2 \cdot 3) \end{array}]$

Step 6.8.4

Rewrite the expression.

$[\begin{array}{c} \frac{7}{34} & - \frac{2}{17} \\ - \frac{1}{17} & \frac{1}{17} \cdot 3 \end{array}]$

Step 6.9

Combine $\frac{1}{17}$ and $3$ .

$[\begin{array}{c} \frac{7}{34} & - \frac{2}{17} \\ - \frac{1}{17} & \frac{3}{17} \end{array}]$