Finite Math Examples

$[\begin{array}{c} 1 & 4 \\ 5 & 7 \end{array}]$

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

$1 \cdot 7 - 5 \cdot 4$

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 $7$ by $1$ .

$7 - 5 \cdot 4$

Step 2.2.1.2

Multiply $- 5$ by $4$ .

$7 - 20$

Step 2.2.2

Subtract $20$ from $7$ .

$- 13$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- 13} [\begin{array}{c} 7 & - 4 \\ - 5 & 1 \end{array}]$

Step 5

Move the negative in front of the fraction.

$- \frac{1}{13} [\begin{array}{c} 7 & - 4 \\ - 5 & 1 \end{array}]$

Step 6

Multiply $- \frac{1}{13}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{13} \cdot 7 & - \frac{1}{13} \cdot - 4 \\ - \frac{1}{13} \cdot - 5 & - \frac{1}{13} \cdot 1 \end{array}]$

Step 7

Simplify each element in the matrix.

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Step 7.1

Multiply $- \frac{1}{13} \cdot 7$ .

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Step 7.1.1

Multiply $7$ by $- 1$ .

$[\begin{array}{c} - 7 (\frac{1}{13}) & - \frac{1}{13} \cdot - 4 \\ - \frac{1}{13} \cdot - 5 & - \frac{1}{13} \cdot 1 \end{array}]$

Step 7.1.2

Combine $- 7$ and $\frac{1}{13}$ .

$[\begin{array}{c} \frac{- 7}{13} & - \frac{1}{13} \cdot - 4 \\ - \frac{1}{13} \cdot - 5 & - \frac{1}{13} \cdot 1 \end{array}]$

Step 7.2

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{7}{13} & - \frac{1}{13} \cdot - 4 \\ - \frac{1}{13} \cdot - 5 & - \frac{1}{13} \cdot 1 \end{array}]$

Step 7.3

Multiply $- \frac{1}{13} \cdot - 4$ .

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Step 7.3.1

Multiply $- 4$ by $- 1$ .

$[\begin{array}{c} - \frac{7}{13} & 4 (\frac{1}{13}) \\ - \frac{1}{13} \cdot - 5 & - \frac{1}{13} \cdot 1 \end{array}]$

Step 7.3.2

Combine $4$ and $\frac{1}{13}$ .

$[\begin{array}{c} - \frac{7}{13} & \frac{4}{13} \\ - \frac{1}{13} \cdot - 5 & - \frac{1}{13} \cdot 1 \end{array}]$

Step 7.4

Multiply $- \frac{1}{13} \cdot - 5$ .

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Step 7.4.1

Multiply $- 5$ by $- 1$ .

$[\begin{array}{c} - \frac{7}{13} & \frac{4}{13} \\ 5 (\frac{1}{13}) & - \frac{1}{13} \cdot 1 \end{array}]$

Step 7.4.2

Combine $5$ and $\frac{1}{13}$ .

$[\begin{array}{c} - \frac{7}{13} & \frac{4}{13} \\ \frac{5}{13} & - \frac{1}{13} \cdot 1 \end{array}]$

Step 7.5

Multiply $- 1$ by $1$ .

$[\begin{array}{c} - \frac{7}{13} & \frac{4}{13} \\ \frac{5}{13} & - \frac{1}{13} \end{array}]$