Finite Math Examples

$[\begin{array}{c} 5 & 1 \\ - 2 & - 2 \end{array}]$

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

$5 \cdot - 2 - (- 2 \cdot 1)$

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 $5$ by $- 2$ .

$- 10 - (- 2 \cdot 1)$

Step 2.2.1.2

Multiply $- (- 2 \cdot 1)$ .

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Step 2.2.1.2.1

Multiply $- 2$ by $1$ .

$- 10 - - 2$

Step 2.2.1.2.2

Multiply $- 1$ by $- 2$ .

$- 10 + 2$

Step 2.2.2

Add $- 10$ and $2$ .

$- 8$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- 8} [\begin{array}{c} - 2 & - 1 \\ 2 & 5 \end{array}]$

Step 5

Move the negative in front of the fraction.

$- \frac{1}{8} [\begin{array}{c} - 2 & - 1 \\ 2 & 5 \end{array}]$

Step 6

Multiply $- \frac{1}{8}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{8} \cdot - 2 & - \frac{1}{8} \cdot - 1 \\ - \frac{1}{8} \cdot 2 & - \frac{1}{8} \cdot 5 \end{array}]$

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 $- \frac{1}{8}$ into the numerator.

$[\begin{array}{c} \frac{- 1}{8} \cdot - 2 & - \frac{1}{8} \cdot - 1 \\ - \frac{1}{8} \cdot 2 & - \frac{1}{8} \cdot 5 \end{array}]$

Step 7.1.2

Factor $2$ out of $8$ .

$[\begin{array}{c} \frac{- 1}{2 (4)} \cdot - 2 & - \frac{1}{8} \cdot - 1 \\ - \frac{1}{8} \cdot 2 & - \frac{1}{8} \cdot 5 \end{array}]$

Step 7.1.3

Factor $2$ out of $- 2$ .

$[\begin{array}{c} \frac{- 1}{2 \cdot 4} \cdot (2 \cdot - 1) & - \frac{1}{8} \cdot - 1 \\ - \frac{1}{8} \cdot 2 & - \frac{1}{8} \cdot 5 \end{array}]$

Step 7.1.4

Cancel the common factor.

$[\begin{array}{c} \frac{- 1}{2 \cdot 4} \cdot (2 \cdot - 1) & - \frac{1}{8} \cdot - 1 \\ - \frac{1}{8} \cdot 2 & - \frac{1}{8} \cdot 5 \end{array}]$

Step 7.1.5

Rewrite the expression.

$[\begin{array}{c} \frac{- 1}{4} \cdot - 1 & - \frac{1}{8} \cdot - 1 \\ - \frac{1}{8} \cdot 2 & - \frac{1}{8} \cdot 5 \end{array}]$

Step 7.2

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

$[\begin{array}{c} \frac{- - 1}{4} & - \frac{1}{8} \cdot - 1 \\ - \frac{1}{8} \cdot 2 & - \frac{1}{8} \cdot 5 \end{array}]$

Step 7.3

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} \frac{1}{4} & - \frac{1}{8} \cdot - 1 \\ - \frac{1}{8} \cdot 2 & - \frac{1}{8} \cdot 5 \end{array}]$

Step 7.4

Multiply $- \frac{1}{8} \cdot - 1$ .

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Step 7.4.1

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} \frac{1}{4} & 1 (\frac{1}{8}) \\ - \frac{1}{8} \cdot 2 & - \frac{1}{8} \cdot 5 \end{array}]$

Step 7.4.2

Multiply $\frac{1}{8}$ by $1$ .

$[\begin{array}{c} \frac{1}{4} & \frac{1}{8} \\ - \frac{1}{8} \cdot 2 & - \frac{1}{8} \cdot 5 \end{array}]$

Step 7.5

Cancel the common factor of $2$ .

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Step 7.5.1

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$[\begin{array}{c} \frac{1}{4} & \frac{1}{8} \\ \frac{- 1}{8} \cdot 2 & - \frac{1}{8} \cdot 5 \end{array}]$

Step 7.5.2

Factor $2$ out of $8$ .

$[\begin{array}{c} \frac{1}{4} & \frac{1}{8} \\ \frac{- 1}{2 (4)} \cdot 2 & - \frac{1}{8} \cdot 5 \end{array}]$

Step 7.5.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{4} & \frac{1}{8} \\ \frac{- 1}{2 \cdot 4} \cdot 2 & - \frac{1}{8} \cdot 5 \end{array}]$

Step 7.5.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{4} & \frac{1}{8} \\ \frac{- 1}{4} & - \frac{1}{8} \cdot 5 \end{array}]$

Step 7.6

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{4} & \frac{1}{8} \\ - \frac{1}{4} & - \frac{1}{8} \cdot 5 \end{array}]$

Step 7.7

Multiply $- \frac{1}{8} \cdot 5$ .

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Step 7.7.1

Multiply $5$ by $- 1$ .

$[\begin{array}{c} \frac{1}{4} & \frac{1}{8} \\ - \frac{1}{4} & - 5 (\frac{1}{8}) \end{array}]$

Step 7.7.2

Combine $- 5$ and $\frac{1}{8}$ .

$[\begin{array}{c} \frac{1}{4} & \frac{1}{8} \\ - \frac{1}{4} & \frac{- 5}{8} \end{array}]$

Step 7.8

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{4} & \frac{1}{8} \\ - \frac{1}{4} & - \frac{5}{8} \end{array}]$