Precalculus Examples

$[\begin{array}{c} - 7 & 5 \\ 12 & 2 \end{array}]$

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

$- 7 \cdot 2 - 12 \cdot 5$

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 $2$ .

$- 14 - 12 \cdot 5$

Step 2.2.1.2

Multiply $- 12$ by $5$ .

$- 14 - 60$

Step 2.2.2

Subtract $60$ from $- 14$ .

$- 74$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- 74} [\begin{array}{c} 2 & - 5 \\ - 12 & - 7 \end{array}]$

Step 5

Move the negative in front of the fraction.

$- \frac{1}{74} [\begin{array}{c} 2 & - 5 \\ - 12 & - 7 \end{array}]$

Step 6

Multiply $- \frac{1}{74}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{74} \cdot 2 & - \frac{1}{74} \cdot - 5 \\ - \frac{1}{74} \cdot - 12 & - \frac{1}{74} \cdot - 7 \end{array}]$

Step 7

Simplify each element in the matrix.

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Step 7.1

Cancel the common factor of $2$ .

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Step 7.1.1

Move the leading negative in $- \frac{1}{74}$ into the numerator.

$[\begin{array}{c} \frac{- 1}{74} \cdot 2 & - \frac{1}{74} \cdot - 5 \\ - \frac{1}{74} \cdot - 12 & - \frac{1}{74} \cdot - 7 \end{array}]$

Step 7.1.2

Factor $2$ out of $74$ .

$[\begin{array}{c} \frac{- 1}{2 (37)} \cdot 2 & - \frac{1}{74} \cdot - 5 \\ - \frac{1}{74} \cdot - 12 & - \frac{1}{74} \cdot - 7 \end{array}]$

Step 7.1.3

Cancel the common factor.

$[\begin{array}{c} \frac{- 1}{2 \cdot 37} \cdot 2 & - \frac{1}{74} \cdot - 5 \\ - \frac{1}{74} \cdot - 12 & - \frac{1}{74} \cdot - 7 \end{array}]$

Step 7.1.4

Rewrite the expression.

$[\begin{array}{c} \frac{- 1}{37} & - \frac{1}{74} \cdot - 5 \\ - \frac{1}{74} \cdot - 12 & - \frac{1}{74} \cdot - 7 \end{array}]$

Step 7.2

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{37} & - \frac{1}{74} \cdot - 5 \\ - \frac{1}{74} \cdot - 12 & - \frac{1}{74} \cdot - 7 \end{array}]$

Step 7.3

Multiply $- \frac{1}{74} \cdot - 5$ .

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Step 7.3.1

Multiply $- 5$ by $- 1$ .

$[\begin{array}{c} - \frac{1}{37} & 5 (\frac{1}{74}) \\ - \frac{1}{74} \cdot - 12 & - \frac{1}{74} \cdot - 7 \end{array}]$

Step 7.3.2

Combine $5$ and $\frac{1}{74}$ .

$[\begin{array}{c} - \frac{1}{37} & \frac{5}{74} \\ - \frac{1}{74} \cdot - 12 & - \frac{1}{74} \cdot - 7 \end{array}]$

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Move the leading negative in $- \frac{1}{74}$ into the numerator.

$[\begin{array}{c} - \frac{1}{37} & \frac{5}{74} \\ \frac{- 1}{74} \cdot - 12 & - \frac{1}{74} \cdot - 7 \end{array}]$

Step 7.4.2

Factor $2$ out of $74$ .

$[\begin{array}{c} - \frac{1}{37} & \frac{5}{74} \\ \frac{- 1}{2 (37)} \cdot - 12 & - \frac{1}{74} \cdot - 7 \end{array}]$

Step 7.4.3

Factor $2$ out of $- 12$ .

$[\begin{array}{c} - \frac{1}{37} & \frac{5}{74} \\ \frac{- 1}{2 \cdot 37} \cdot (2 \cdot - 6) & - \frac{1}{74} \cdot - 7 \end{array}]$

Step 7.4.4

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{37} & \frac{5}{74} \\ \frac{- 1}{2 \cdot 37} \cdot (2 \cdot - 6) & - \frac{1}{74} \cdot - 7 \end{array}]$

Step 7.4.5

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{37} & \frac{5}{74} \\ \frac{- 1}{37} \cdot - 6 & - \frac{1}{74} \cdot - 7 \end{array}]$

Step 7.5

Combine $\frac{- 1}{37}$ and $- 6$ .

$[\begin{array}{c} - \frac{1}{37} & \frac{5}{74} \\ \frac{- - 6}{37} & - \frac{1}{74} \cdot - 7 \end{array}]$

Step 7.6

Multiply $- 1$ by $- 6$ .

$[\begin{array}{c} - \frac{1}{37} & \frac{5}{74} \\ \frac{6}{37} & - \frac{1}{74} \cdot - 7 \end{array}]$

Step 7.7

Multiply $- \frac{1}{74} \cdot - 7$ .

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Step 7.7.1

Multiply $- 7$ by $- 1$ .

$[\begin{array}{c} - \frac{1}{37} & \frac{5}{74} \\ \frac{6}{37} & 7 (\frac{1}{74}) \end{array}]$

Step 7.7.2

Combine $7$ and $\frac{1}{74}$ .

$[\begin{array}{c} - \frac{1}{37} & \frac{5}{74} \\ \frac{6}{37} & \frac{7}{74} \end{array}]$