Finite Math Examples

$[\begin{array}{c} 2 & - 4 \\ 6 & 10 \end{array}]$

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

$2 \cdot 10 - 6 \cdot - 4$

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 $2$ by $10$ .

$20 - 6 \cdot - 4$

Step 2.2.1.2

Multiply $- 6$ by $- 4$ .

$20 + 24$

Step 2.2.2

Add $20$ and $24$ .

$44$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{44} [\begin{array}{c} 10 & 4 \\ - 6 & 2 \end{array}]$

Step 5

Multiply $\frac{1}{44}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{44} \cdot 10 & \frac{1}{44} \cdot 4 \\ \frac{1}{44} \cdot - 6 & \frac{1}{44} \cdot 2 \end{array}]$

Step 6

Simplify each element in the matrix.

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Step 6.1

Cancel the common factor of $2$ .

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Step 6.1.1

Factor $2$ out of $44$ .

$[\begin{array}{c} \frac{1}{2 (22)} \cdot 10 & \frac{1}{44} \cdot 4 \\ \frac{1}{44} \cdot - 6 & \frac{1}{44} \cdot 2 \end{array}]$

Step 6.1.2

Factor $2$ out of $10$ .

$[\begin{array}{c} \frac{1}{2 \cdot 22} \cdot (2 \cdot 5) & \frac{1}{44} \cdot 4 \\ \frac{1}{44} \cdot - 6 & \frac{1}{44} \cdot 2 \end{array}]$

Step 6.1.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{2 \cdot 22} \cdot (2 \cdot 5) & \frac{1}{44} \cdot 4 \\ \frac{1}{44} \cdot - 6 & \frac{1}{44} \cdot 2 \end{array}]$

Step 6.1.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{22} \cdot 5 & \frac{1}{44} \cdot 4 \\ \frac{1}{44} \cdot - 6 & \frac{1}{44} \cdot 2 \end{array}]$

Step 6.2

Combine $\frac{1}{22}$ and $5$ .

$[\begin{array}{c} \frac{5}{22} & \frac{1}{44} \cdot 4 \\ \frac{1}{44} \cdot - 6 & \frac{1}{44} \cdot 2 \end{array}]$

Step 6.3

Cancel the common factor of $4$ .

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Step 6.3.1

Factor $4$ out of $44$ .

$[\begin{array}{c} \frac{5}{22} & \frac{1}{4 (11)} \cdot 4 \\ \frac{1}{44} \cdot - 6 & \frac{1}{44} \cdot 2 \end{array}]$

Step 6.3.2

Cancel the common factor.

$[\begin{array}{c} \frac{5}{22} & \frac{1}{4 \cdot 11} \cdot 4 \\ \frac{1}{44} \cdot - 6 & \frac{1}{44} \cdot 2 \end{array}]$

Step 6.3.3

Rewrite the expression.

$[\begin{array}{c} \frac{5}{22} & \frac{1}{11} \\ \frac{1}{44} \cdot - 6 & \frac{1}{44} \cdot 2 \end{array}]$

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Factor $2$ out of $44$ .

$[\begin{array}{c} \frac{5}{22} & \frac{1}{11} \\ \frac{1}{2 (22)} \cdot - 6 & \frac{1}{44} \cdot 2 \end{array}]$

Step 6.4.2

Factor $2$ out of $- 6$ .

$[\begin{array}{c} \frac{5}{22} & \frac{1}{11} \\ \frac{1}{2 \cdot 22} \cdot (2 \cdot - 3) & \frac{1}{44} \cdot 2 \end{array}]$

Step 6.4.3

Cancel the common factor.

$[\begin{array}{c} \frac{5}{22} & \frac{1}{11} \\ \frac{1}{2 \cdot 22} \cdot (2 \cdot - 3) & \frac{1}{44} \cdot 2 \end{array}]$

Step 6.4.4

Rewrite the expression.

$[\begin{array}{c} \frac{5}{22} & \frac{1}{11} \\ \frac{1}{22} \cdot - 3 & \frac{1}{44} \cdot 2 \end{array}]$

Step 6.5

Combine $\frac{1}{22}$ and $- 3$ .

$[\begin{array}{c} \frac{5}{22} & \frac{1}{11} \\ \frac{- 3}{22} & \frac{1}{44} \cdot 2 \end{array}]$

Step 6.6

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{5}{22} & \frac{1}{11} \\ - \frac{3}{22} & \frac{1}{44} \cdot 2 \end{array}]$

Step 6.7

Cancel the common factor of $2$ .

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Step 6.7.1

Factor $2$ out of $44$ .

$[\begin{array}{c} \frac{5}{22} & \frac{1}{11} \\ - \frac{3}{22} & \frac{1}{2 (22)} \cdot 2 \end{array}]$

Step 6.7.2

Cancel the common factor.

$[\begin{array}{c} \frac{5}{22} & \frac{1}{11} \\ - \frac{3}{22} & \frac{1}{2 \cdot 22} \cdot 2 \end{array}]$

Step 6.7.3

Rewrite the expression.

$[\begin{array}{c} \frac{5}{22} & \frac{1}{11} \\ - \frac{3}{22} & \frac{1}{22} \end{array}]$