Finite Math Examples

$4 A = [\begin{array}{c} 9 & 4 \\ 7 & 8 \end{array}]$

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

$9 \cdot 8 - 7 \cdot 4$

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 $9$ by $8$ .

$72 - 7 \cdot 4$

Step 2.2.1.2

Multiply $- 7$ by $4$ .

$72 - 28$

Step 2.2.2

Subtract $28$ from $72$ .

$44$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{44} [\begin{array}{c} 8 & - 4 \\ - 7 & 9 \end{array}]$

Step 5

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

$[\begin{array}{c} \frac{1}{44} \cdot 8 & \frac{1}{44} \cdot - 4 \\ \frac{1}{44} \cdot - 7 & \frac{1}{44} \cdot 9 \end{array}]$

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 $44$ .

$[\begin{array}{c} \frac{1}{4 (11)} \cdot 8 & \frac{1}{44} \cdot - 4 \\ \frac{1}{44} \cdot - 7 & \frac{1}{44} \cdot 9 \end{array}]$

Step 6.1.2

Factor $4$ out of $8$ .

$[\begin{array}{c} \frac{1}{4 \cdot 11} \cdot (4 \cdot 2) & \frac{1}{44} \cdot - 4 \\ \frac{1}{44} \cdot - 7 & \frac{1}{44} \cdot 9 \end{array}]$

Step 6.1.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{4 \cdot 11} \cdot (4 \cdot 2) & \frac{1}{44} \cdot - 4 \\ \frac{1}{44} \cdot - 7 & \frac{1}{44} \cdot 9 \end{array}]$

Step 6.1.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{11} \cdot 2 & \frac{1}{44} \cdot - 4 \\ \frac{1}{44} \cdot - 7 & \frac{1}{44} \cdot 9 \end{array}]$

Step 6.2

Combine $\frac{1}{11}$ and $2$ .

$[\begin{array}{c} \frac{2}{11} & \frac{1}{44} \cdot - 4 \\ \frac{1}{44} \cdot - 7 & \frac{1}{44} \cdot 9 \end{array}]$

Step 6.3

Cancel the common factor of $4$ .

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Step 6.3.1

Factor $4$ out of $44$ .

$[\begin{array}{c} \frac{2}{11} & \frac{1}{4 (11)} \cdot - 4 \\ \frac{1}{44} \cdot - 7 & \frac{1}{44} \cdot 9 \end{array}]$

Step 6.3.2

Factor $4$ out of $- 4$ .

$[\begin{array}{c} \frac{2}{11} & \frac{1}{4 \cdot 11} \cdot (4 \cdot - 1) \\ \frac{1}{44} \cdot - 7 & \frac{1}{44} \cdot 9 \end{array}]$

Step 6.3.3

Cancel the common factor.

$[\begin{array}{c} \frac{2}{11} & \frac{1}{4 \cdot 11} \cdot (4 \cdot - 1) \\ \frac{1}{44} \cdot - 7 & \frac{1}{44} \cdot 9 \end{array}]$

Step 6.3.4

Rewrite the expression.

$[\begin{array}{c} \frac{2}{11} & \frac{1}{11} \cdot - 1 \\ \frac{1}{44} \cdot - 7 & \frac{1}{44} \cdot 9 \end{array}]$

Step 6.4

Combine $\frac{1}{11}$ and $- 1$ .

$[\begin{array}{c} \frac{2}{11} & \frac{- 1}{11} \\ \frac{1}{44} \cdot - 7 & \frac{1}{44} \cdot 9 \end{array}]$

Step 6.5

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{2}{11} & - \frac{1}{11} \\ \frac{1}{44} \cdot - 7 & \frac{1}{44} \cdot 9 \end{array}]$

Step 6.6

Combine $\frac{1}{44}$ and $- 7$ .

$[\begin{array}{c} \frac{2}{11} & - \frac{1}{11} \\ \frac{- 7}{44} & \frac{1}{44} \cdot 9 \end{array}]$

Step 6.7

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{2}{11} & - \frac{1}{11} \\ - \frac{7}{44} & \frac{1}{44} \cdot 9 \end{array}]$

Step 6.8

Combine $\frac{1}{44}$ and $9$ .

$[\begin{array}{c} \frac{2}{11} & - \frac{1}{11} \\ - \frac{7}{44} & \frac{9}{44} \end{array}]$