Precalculus Examples

$[\begin{array}{c} 8 & - 4 \\ 2 & 1 \end{array}]$

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

$8 \cdot 1 - 2 \cdot - 4$

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 $8$ by $1$ .

$8 - 2 \cdot - 4$

Step 2.2.1.2

Multiply $- 2$ by $- 4$ .

$8 + 8$

Step 2.2.2

Add $8$ and $8$ .

$16$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{16} [\begin{array}{c} 1 & 4 \\ - 2 & 8 \end{array}]$

Step 5

Multiply $\frac{1}{16}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{16} \cdot 1 & \frac{1}{16} \cdot 4 \\ \frac{1}{16} \cdot - 2 & \frac{1}{16} \cdot 8 \end{array}]$

Step 6

Simplify each element in the matrix.

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Step 6.1

Multiply $\frac{1}{16}$ by $1$ .

$[\begin{array}{c} \frac{1}{16} & \frac{1}{16} \cdot 4 \\ \frac{1}{16} \cdot - 2 & \frac{1}{16} \cdot 8 \end{array}]$

Step 6.2

Cancel the common factor of $4$ .

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Step 6.2.1

Factor $4$ out of $16$ .

$[\begin{array}{c} \frac{1}{16} & \frac{1}{4 (4)} \cdot 4 \\ \frac{1}{16} \cdot - 2 & \frac{1}{16} \cdot 8 \end{array}]$

Step 6.2.2

Cancel the common factor.

$[\begin{array}{c} \frac{1}{16} & \frac{1}{4 \cdot 4} \cdot 4 \\ \frac{1}{16} \cdot - 2 & \frac{1}{16} \cdot 8 \end{array}]$

Step 6.2.3

Rewrite the expression.

$[\begin{array}{c} \frac{1}{16} & \frac{1}{4} \\ \frac{1}{16} \cdot - 2 & \frac{1}{16} \cdot 8 \end{array}]$

Step 6.3

Cancel the common factor of $2$ .

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Step 6.3.1

Factor $2$ out of $16$ .

$[\begin{array}{c} \frac{1}{16} & \frac{1}{4} \\ \frac{1}{2 (8)} \cdot - 2 & \frac{1}{16} \cdot 8 \end{array}]$

Step 6.3.2

Factor $2$ out of $- 2$ .

$[\begin{array}{c} \frac{1}{16} & \frac{1}{4} \\ \frac{1}{2 \cdot 8} \cdot (2 \cdot - 1) & \frac{1}{16} \cdot 8 \end{array}]$

Step 6.3.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{16} & \frac{1}{4} \\ \frac{1}{2 \cdot 8} \cdot (2 \cdot - 1) & \frac{1}{16} \cdot 8 \end{array}]$

Step 6.3.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{16} & \frac{1}{4} \\ \frac{1}{8} \cdot - 1 & \frac{1}{16} \cdot 8 \end{array}]$

Step 6.4

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

$[\begin{array}{c} \frac{1}{16} & \frac{1}{4} \\ \frac{- 1}{8} & \frac{1}{16} \cdot 8 \end{array}]$

Step 6.5

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{16} & \frac{1}{4} \\ - \frac{1}{8} & \frac{1}{16} \cdot 8 \end{array}]$

Step 6.6

Cancel the common factor of $8$ .

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Step 6.6.1

Factor $8$ out of $16$ .

$[\begin{array}{c} \frac{1}{16} & \frac{1}{4} \\ - \frac{1}{8} & \frac{1}{8 (2)} \cdot 8 \end{array}]$

Step 6.6.2

Cancel the common factor.

$[\begin{array}{c} \frac{1}{16} & \frac{1}{4} \\ - \frac{1}{8} & \frac{1}{8 \cdot 2} \cdot 8 \end{array}]$

Step 6.6.3

Rewrite the expression.

$[\begin{array}{c} \frac{1}{16} & \frac{1}{4} \\ - \frac{1}{8} & \frac{1}{2} \end{array}]$