Finite Math Examples

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

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

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

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 $4$ .

$- 8 - 1 \cdot 8$

Step 2.2.1.2

Multiply $- 1$ by $8$ .

$- 8 - 8$

Step 2.2.2

Subtract $8$ from $- 8$ .

$- 16$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

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

Step 5

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

Simplify each element in the matrix.

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Step 7.1

Cancel the common factor of $4$ .

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Step 7.1.1

Move the leading negative in $- \frac{1}{16}$ into the numerator.

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

Step 7.1.2

Factor $4$ out of $16$ .

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

Step 7.1.3

Cancel the common factor.

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

Step 7.1.4

Rewrite the expression.

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

Step 7.2

Move the negative in front of the fraction.

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

Step 7.3

Cancel the common factor of $8$ .

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Step 7.3.1

Move the leading negative in $- \frac{1}{16}$ into the numerator.

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

Step 7.3.2

Factor $8$ out of $16$ .

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

Step 7.3.3

Factor $8$ out of $- 8$ .

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

Step 7.3.4

Cancel the common factor.

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

Step 7.3.5

Rewrite the expression.

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

Step 7.4

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

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

Step 7.5

Multiply $- 1$ by $- 1$ .

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

Step 7.6

Multiply $- \frac{1}{16} \cdot - 1$ .

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Step 7.6.1

Multiply $- 1$ by $- 1$ .

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

Step 7.6.2

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

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

Step 7.7

Cancel the common factor of $2$ .

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Step 7.7.1

Move the leading negative in $- \frac{1}{16}$ into the numerator.

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

Step 7.7.2

Factor $2$ out of $16$ .

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

Step 7.7.3

Factor $2$ out of $- 2$ .

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

Step 7.7.4

Cancel the common factor.

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

Step 7.7.5

Rewrite the expression.

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

Step 7.8

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

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

Step 7.9

Multiply $- 1$ by $- 1$ .

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