Finite Math Examples

$[\begin{array}{c} 2 & - 4 \\ - 4 & 9 \end{array}]$

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

$2 \cdot 9 - (- 4 \cdot - 4)$

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 $9$ .

$18 - (- 4 \cdot - 4)$

Step 2.2.1.2

Multiply $- (- 4 \cdot - 4)$ .

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Step 2.2.1.2.1

Multiply $- 4$ by $- 4$ .

$18 - 1 \cdot 16$

Step 2.2.1.2.2

Multiply $- 1$ by $16$ .

$18 - 16$

Step 2.2.2

Subtract $16$ from $18$ .

$2$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{2} [\begin{array}{c} 9 & 4 \\ 4 & 2 \end{array}]$

Step 5

Multiply $\frac{1}{2}$ by each element of the matrix.

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

Step 6

Simplify each element in the matrix.

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Step 6.1

Combine $\frac{1}{2}$ and $9$ .

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

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Factor $2$ out of $4$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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

Step 6.3

Cancel the common factor of $2$ .

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Step 6.3.1

Factor $2$ out of $4$ .

$[\begin{array}{c} \frac{9}{2} & 2 \\ \frac{1}{2} \cdot (2 (2)) & \frac{1}{2} \cdot 2 \end{array}]$

Step 6.3.2

Cancel the common factor.

$[\begin{array}{c} \frac{9}{2} & 2 \\ \frac{1}{2} \cdot (2 \cdot 2) & \frac{1}{2} \cdot 2 \end{array}]$

Step 6.3.3

Rewrite the expression.

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

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Cancel the common factor.

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

Step 6.4.2

Rewrite the expression.

$[\begin{array}{c} \frac{9}{2} & 2 \\ 2 & 1 \end{array}]$