Precalculus Examples

$[\begin{array}{c} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ - \frac{\sqrt{3}}{2} & \frac{1}{2} \end{array}]$

Step 1

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

$\frac{1}{2} \cdot \frac{1}{2} - (- \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2})$

Step 2

Simplify the determinant.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 2.1.1.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\frac{1}{2 \cdot 2} - (- \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2})$

Step 2.1.1.2

Multiply $2$ by $2$ .

$\frac{1}{4} - (- \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2})$

Step 2.1.2

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

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Step 2.1.2.1

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$\frac{1}{4} - - \frac{\sqrt{3} \sqrt{3}}{2 \cdot 2}$

Step 2.1.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{1}{4} - - \frac{{\sqrt{3}}^{1} \sqrt{3}}{2 \cdot 2}$

Step 2.1.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{1}{4} - - \frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{2 \cdot 2}$

Step 2.1.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{4} - - \frac{{\sqrt{3}}^{1 + 1}}{2 \cdot 2}$

Step 2.1.2.5

Add $1$ and $1$ .

$\frac{1}{4} - - \frac{{\sqrt{3}}^{2}}{2 \cdot 2}$

Step 2.1.2.6

Multiply $2$ by $2$ .

$\frac{1}{4} - - \frac{{\sqrt{3}}^{2}}{4}$

Step 2.1.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{1}{4} - - \frac{{(3^{\frac{1}{2}})}^{2}}{4}$

Step 2.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{4} - - \frac{3^{\frac{1}{2} \cdot 2}}{4}$

Step 2.1.3.3

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

$\frac{1}{4} - - \frac{3^{\frac{2}{2}}}{4}$

Step 2.1.3.4

Cancel the common factor of $2$ .

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Step 2.1.3.4.1

Cancel the common factor.

$\frac{1}{4} - - \frac{3^{\frac{2}{2}}}{4}$

Step 2.1.3.4.2

Rewrite the expression.

$\frac{1}{4} - - \frac{3^{1}}{4}$

Step 2.1.3.5

Evaluate the exponent.

$\frac{1}{4} - - \frac{3}{4}$

Step 2.1.4

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

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Step 2.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{4} + 1 (\frac{3}{4})$

Step 2.1.4.2

Multiply $\frac{3}{4}$ by $1$ .

$\frac{1}{4} + \frac{3}{4}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{1 + 3}{4}$

Step 2.3

Add $1$ and $3$ .

$\frac{4}{4}$

Step 2.4

Divide $4$ by $4$ .

$1$