Algebra Examples

$[\begin{array}{c} \frac{1}{3} & \frac{1}{3} \\ \frac{1}{12} & - \frac{5}{6} \end{array}]$

Step 1

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

$\frac{1}{3} (- \frac{5}{6}) - \frac{1}{12} \cdot \frac{1}{3}$

Step 2

Simplify the determinant.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $\frac{1}{3} (- \frac{5}{6})$ .

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Step 2.1.1.1

Multiply $\frac{1}{3}$ by $\frac{5}{6}$ .

$- \frac{5}{3 \cdot 6} - \frac{1}{12} \cdot \frac{1}{3}$

Step 2.1.1.2

Multiply $3$ by $6$ .

$- \frac{5}{18} - \frac{1}{12} \cdot \frac{1}{3}$

Step 2.1.2

Multiply $- \frac{1}{12} \cdot \frac{1}{3}$ .

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Step 2.1.2.1

Multiply $\frac{1}{3}$ by $\frac{1}{12}$ .

$- \frac{5}{18} - \frac{1}{3 \cdot 12}$

Step 2.1.2.2

Multiply $3$ by $12$ .

$- \frac{5}{18} - \frac{1}{36}$

Step 2.2

To write $- \frac{5}{18}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{5}{18} \cdot \frac{2}{2} - \frac{1}{36}$

Step 2.3

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{5}{18}$ by $\frac{2}{2}$ .

$- \frac{5 \cdot 2}{18 \cdot 2} - \frac{1}{36}$

Step 2.3.2

Multiply $18$ by $2$ .

$- \frac{5 \cdot 2}{36} - \frac{1}{36}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{- 5 \cdot 2 - 1}{36}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 5$ by $2$ .

$\frac{- 10 - 1}{36}$

Step 2.5.2

Subtract $1$ from $- 10$ .

$\frac{- 11}{36}$

Step 2.6

Move the negative in front of the fraction.

$- \frac{11}{36}$