Precalculus Examples

$[\begin{array}{c} 2 & 2 \\ 3 & - 7 \end{array}]$

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

$2 \cdot - 7 - 3 \cdot 2$

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 $2$ by $- 7$ .

$- 14 - 3 \cdot 2$

Step 2.2.1.2

Multiply $- 3$ by $2$ .

$- 14 - 6$

Step 2.2.2

Subtract $6$ from $- 14$ .

$- 20$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- 20} [\begin{array}{c} - 7 & - 2 \\ - 3 & 2 \end{array}]$

Step 5

Move the negative in front of the fraction.

$- \frac{1}{20} [\begin{array}{c} - 7 & - 2 \\ - 3 & 2 \end{array}]$

Step 6

Multiply $- \frac{1}{20}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{20} \cdot - 7 & - \frac{1}{20} \cdot - 2 \\ - \frac{1}{20} \cdot - 3 & - \frac{1}{20} \cdot 2 \end{array}]$

Step 7

Simplify each element in the matrix.

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Step 7.1

Multiply $- \frac{1}{20} \cdot - 7$ .

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Step 7.1.1

Multiply $- 7$ by $- 1$ .

$[\begin{array}{c} 7 (\frac{1}{20}) & - \frac{1}{20} \cdot - 2 \\ - \frac{1}{20} \cdot - 3 & - \frac{1}{20} \cdot 2 \end{array}]$

Step 7.1.2

Combine $7$ and $\frac{1}{20}$ .

$[\begin{array}{c} \frac{7}{20} & - \frac{1}{20} \cdot - 2 \\ - \frac{1}{20} \cdot - 3 & - \frac{1}{20} \cdot 2 \end{array}]$

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Move the leading negative in $- \frac{1}{20}$ into the numerator.

$[\begin{array}{c} \frac{7}{20} & \frac{- 1}{20} \cdot - 2 \\ - \frac{1}{20} \cdot - 3 & - \frac{1}{20} \cdot 2 \end{array}]$

Step 7.2.2

Factor $2$ out of $20$ .

$[\begin{array}{c} \frac{7}{20} & \frac{- 1}{2 (10)} \cdot - 2 \\ - \frac{1}{20} \cdot - 3 & - \frac{1}{20} \cdot 2 \end{array}]$

Step 7.2.3

Factor $2$ out of $- 2$ .

$[\begin{array}{c} \frac{7}{20} & \frac{- 1}{2 \cdot 10} \cdot (2 \cdot - 1) \\ - \frac{1}{20} \cdot - 3 & - \frac{1}{20} \cdot 2 \end{array}]$

Step 7.2.4

Cancel the common factor.

$[\begin{array}{c} \frac{7}{20} & \frac{- 1}{2 \cdot 10} \cdot (2 \cdot - 1) \\ - \frac{1}{20} \cdot - 3 & - \frac{1}{20} \cdot 2 \end{array}]$

Step 7.2.5

Rewrite the expression.

$[\begin{array}{c} \frac{7}{20} & \frac{- 1}{10} \cdot - 1 \\ - \frac{1}{20} \cdot - 3 & - \frac{1}{20} \cdot 2 \end{array}]$

Step 7.3

Combine $\frac{- 1}{10}$ and $- 1$ .

$[\begin{array}{c} \frac{7}{20} & \frac{- - 1}{10} \\ - \frac{1}{20} \cdot - 3 & - \frac{1}{20} \cdot 2 \end{array}]$

Step 7.4

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} \frac{7}{20} & \frac{1}{10} \\ - \frac{1}{20} \cdot - 3 & - \frac{1}{20} \cdot 2 \end{array}]$

Step 7.5

Multiply $- \frac{1}{20} \cdot - 3$ .

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Step 7.5.1

Multiply $- 3$ by $- 1$ .

$[\begin{array}{c} \frac{7}{20} & \frac{1}{10} \\ 3 (\frac{1}{20}) & - \frac{1}{20} \cdot 2 \end{array}]$

Step 7.5.2

Combine $3$ and $\frac{1}{20}$ .

$[\begin{array}{c} \frac{7}{20} & \frac{1}{10} \\ \frac{3}{20} & - \frac{1}{20} \cdot 2 \end{array}]$

Step 7.6

Cancel the common factor of $2$ .

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Step 7.6.1

Move the leading negative in $- \frac{1}{20}$ into the numerator.

$[\begin{array}{c} \frac{7}{20} & \frac{1}{10} \\ \frac{3}{20} & \frac{- 1}{20} \cdot 2 \end{array}]$

Step 7.6.2

Factor $2$ out of $20$ .

$[\begin{array}{c} \frac{7}{20} & \frac{1}{10} \\ \frac{3}{20} & \frac{- 1}{2 (10)} \cdot 2 \end{array}]$

Step 7.6.3

Cancel the common factor.

$[\begin{array}{c} \frac{7}{20} & \frac{1}{10} \\ \frac{3}{20} & \frac{- 1}{2 \cdot 10} \cdot 2 \end{array}]$

Step 7.6.4

Rewrite the expression.

$[\begin{array}{c} \frac{7}{20} & \frac{1}{10} \\ \frac{3}{20} & \frac{- 1}{10} \end{array}]$

Step 7.7

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{7}{20} & \frac{1}{10} \\ \frac{3}{20} & - \frac{1}{10} \end{array}]$