Finite Math Examples

$[\begin{array}{c} 18 & - 32 \\ - 9 & 20 \end{array}]$

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

$18 \cdot 20 - (- 9 \cdot - 32)$

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 $18$ by $20$ .

$360 - (- 9 \cdot - 32)$

Step 2.2.1.2

Multiply $- (- 9 \cdot - 32)$ .

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Step 2.2.1.2.1

Multiply $- 9$ by $- 32$ .

$360 - 1 \cdot 288$

Step 2.2.1.2.2

Multiply $- 1$ by $288$ .

$360 - 288$

Step 2.2.2

Subtract $288$ from $360$ .

$72$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{72} [\begin{array}{c} 20 & 32 \\ 9 & 18 \end{array}]$

Step 5

Multiply $\frac{1}{72}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{72} \cdot 20 & \frac{1}{72} \cdot 32 \\ \frac{1}{72} \cdot 9 & \frac{1}{72} \cdot 18 \end{array}]$

Step 6

Simplify each element in the matrix.

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Step 6.1

Cancel the common factor of $4$ .

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Step 6.1.1

Factor $4$ out of $72$ .

$[\begin{array}{c} \frac{1}{4 (18)} \cdot 20 & \frac{1}{72} \cdot 32 \\ \frac{1}{72} \cdot 9 & \frac{1}{72} \cdot 18 \end{array}]$

Step 6.1.2

Factor $4$ out of $20$ .

$[\begin{array}{c} \frac{1}{4 \cdot 18} \cdot (4 \cdot 5) & \frac{1}{72} \cdot 32 \\ \frac{1}{72} \cdot 9 & \frac{1}{72} \cdot 18 \end{array}]$

Step 6.1.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{4 \cdot 18} \cdot (4 \cdot 5) & \frac{1}{72} \cdot 32 \\ \frac{1}{72} \cdot 9 & \frac{1}{72} \cdot 18 \end{array}]$

Step 6.1.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{18} \cdot 5 & \frac{1}{72} \cdot 32 \\ \frac{1}{72} \cdot 9 & \frac{1}{72} \cdot 18 \end{array}]$

Step 6.2

Combine $\frac{1}{18}$ and $5$ .

$[\begin{array}{c} \frac{5}{18} & \frac{1}{72} \cdot 32 \\ \frac{1}{72} \cdot 9 & \frac{1}{72} \cdot 18 \end{array}]$

Step 6.3

Cancel the common factor of $8$ .

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Step 6.3.1

Factor $8$ out of $72$ .

$[\begin{array}{c} \frac{5}{18} & \frac{1}{8 (9)} \cdot 32 \\ \frac{1}{72} \cdot 9 & \frac{1}{72} \cdot 18 \end{array}]$

Step 6.3.2

Factor $8$ out of $32$ .

$[\begin{array}{c} \frac{5}{18} & \frac{1}{8 \cdot 9} \cdot (8 \cdot 4) \\ \frac{1}{72} \cdot 9 & \frac{1}{72} \cdot 18 \end{array}]$

Step 6.3.3

Cancel the common factor.

$[\begin{array}{c} \frac{5}{18} & \frac{1}{8 \cdot 9} \cdot (8 \cdot 4) \\ \frac{1}{72} \cdot 9 & \frac{1}{72} \cdot 18 \end{array}]$

Step 6.3.4

Rewrite the expression.

$[\begin{array}{c} \frac{5}{18} & \frac{1}{9} \cdot 4 \\ \frac{1}{72} \cdot 9 & \frac{1}{72} \cdot 18 \end{array}]$

Step 6.4

Combine $\frac{1}{9}$ and $4$ .

$[\begin{array}{c} \frac{5}{18} & \frac{4}{9} \\ \frac{1}{72} \cdot 9 & \frac{1}{72} \cdot 18 \end{array}]$

Step 6.5

Cancel the common factor of $9$ .

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Step 6.5.1

Factor $9$ out of $72$ .

$[\begin{array}{c} \frac{5}{18} & \frac{4}{9} \\ \frac{1}{9 (8)} \cdot 9 & \frac{1}{72} \cdot 18 \end{array}]$

Step 6.5.2

Cancel the common factor.

$[\begin{array}{c} \frac{5}{18} & \frac{4}{9} \\ \frac{1}{9 \cdot 8} \cdot 9 & \frac{1}{72} \cdot 18 \end{array}]$

Step 6.5.3

Rewrite the expression.

$[\begin{array}{c} \frac{5}{18} & \frac{4}{9} \\ \frac{1}{8} & \frac{1}{72} \cdot 18 \end{array}]$

Step 6.6

Cancel the common factor of $18$ .

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Step 6.6.1

Factor $18$ out of $72$ .

$[\begin{array}{c} \frac{5}{18} & \frac{4}{9} \\ \frac{1}{8} & \frac{1}{18 (4)} \cdot 18 \end{array}]$

Step 6.6.2

Cancel the common factor.

$[\begin{array}{c} \frac{5}{18} & \frac{4}{9} \\ \frac{1}{8} & \frac{1}{18 \cdot 4} \cdot 18 \end{array}]$

Step 6.6.3

Rewrite the expression.

$[\begin{array}{c} \frac{5}{18} & \frac{4}{9} \\ \frac{1}{8} & \frac{1}{4} \end{array}]$