Precalculus Examples

$[\begin{array}{c} \frac{4}{5} & 4 \\ \frac{2}{3} & 5 \end{array}]$

Step 1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$\frac{4}{5} \cdot 5 - \frac{2}{3} \cdot 4$

Step 2

Simplify the determinant.

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Step 2.1

Simplify each term.

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Step 2.1.1

Cancel the common factor of $5$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{4}{5} \cdot 5 - \frac{2}{3} \cdot 4$

Step 2.1.1.2

Rewrite the expression.

$4 - \frac{2}{3} \cdot 4$

Step 2.1.2

Multiply $- \frac{2}{3} \cdot 4$ .

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Step 2.1.2.1

Multiply $4$ by $- 1$ .

$4 - 4 (\frac{2}{3})$

Step 2.1.2.2

Combine $- 4$ and $\frac{2}{3}$ .

$4 + \frac{- 4 \cdot 2}{3}$

Step 2.1.2.3

Multiply $- 4$ by $2$ .

$4 + \frac{- 8}{3}$

Step 2.1.3

Move the negative in front of the fraction.

$4 - \frac{8}{3}$

Step 2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4 \cdot \frac{3}{3} - \frac{8}{3}$

Step 2.3

Combine $4$ and $\frac{3}{3}$ .

$\frac{4 \cdot 3}{3} - \frac{8}{3}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{4 \cdot 3 - 8}{3}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $4$ by $3$ .

$\frac{12 - 8}{3}$

Step 2.5.2

Subtract $8$ from $12$ .

$\frac{4}{3}$