Algebra Examples

$[\begin{array}{c} \frac{1}{7} & \frac{1}{10} \\ - \frac{6}{5} & \frac{10}{9} \end{array}]$

Step 1

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

$\frac{1}{7} \cdot \frac{10}{9} - (- \frac{6}{5} \cdot \frac{1}{10})$

Step 2

Simplify the determinant.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $\frac{1}{7} \cdot \frac{10}{9}$ .

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Step 2.1.1.1

Multiply $\frac{1}{7}$ by $\frac{10}{9}$ .

$\frac{10}{7 \cdot 9} - (- \frac{6}{5} \cdot \frac{1}{10})$

Step 2.1.1.2

Multiply $7$ by $9$ .

$\frac{10}{63} - (- \frac{6}{5} \cdot \frac{1}{10})$

Step 2.1.2

Cancel the common factor of $2$ .

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Step 2.1.2.1

Move the leading negative in $- \frac{6}{5}$ into the numerator.

$\frac{10}{63} - (\frac{- 6}{5} \cdot \frac{1}{10})$

Step 2.1.2.2

Factor $2$ out of $- 6$ .

$\frac{10}{63} - (\frac{2 (- 3)}{5} \cdot \frac{1}{10})$

Step 2.1.2.3

Factor $2$ out of $10$ .

$\frac{10}{63} - (\frac{2 \cdot - 3}{5} \cdot \frac{1}{2 \cdot 5})$

Step 2.1.2.4

Cancel the common factor.

$\frac{10}{63} - (\frac{2 \cdot - 3}{5} \cdot \frac{1}{2 \cdot 5})$

Step 2.1.2.5

Rewrite the expression.

$\frac{10}{63} - (\frac{- 3}{5} \cdot \frac{1}{5})$

Step 2.1.3

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

$\frac{10}{63} - \frac{- 3}{5 \cdot 5}$

Step 2.1.4

Multiply $5$ by $5$ .

$\frac{10}{63} - \frac{- 3}{25}$

Step 2.1.5

Move the negative in front of the fraction.

$\frac{10}{63} - - \frac{3}{25}$

Step 2.1.6

Multiply $- - \frac{3}{25}$ .

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Step 2.1.6.1

Multiply $- 1$ by $- 1$ .

$\frac{10}{63} + 1 (\frac{3}{25})$

Step 2.1.6.2

Multiply $\frac{3}{25}$ by $1$ .

$\frac{10}{63} + \frac{3}{25}$

Step 2.2

To write $\frac{10}{63}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\frac{10}{63} \cdot \frac{25}{25} + \frac{3}{25}$

Step 2.3

To write $\frac{3}{25}$ as a fraction with a common denominator, multiply by $\frac{63}{63}$ .

$\frac{10}{63} \cdot \frac{25}{25} + \frac{3}{25} \cdot \frac{63}{63}$

Step 2.4

Write each expression with a common denominator of $1575$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{10}{63}$ by $\frac{25}{25}$ .

$\frac{10 \cdot 25}{63 \cdot 25} + \frac{3}{25} \cdot \frac{63}{63}$

Step 2.4.2

Multiply $63$ by $25$ .

$\frac{10 \cdot 25}{1575} + \frac{3}{25} \cdot \frac{63}{63}$

Step 2.4.3

Multiply $\frac{3}{25}$ by $\frac{63}{63}$ .

$\frac{10 \cdot 25}{1575} + \frac{3 \cdot 63}{25 \cdot 63}$

Step 2.4.4

Multiply $25$ by $63$ .

$\frac{10 \cdot 25}{1575} + \frac{3 \cdot 63}{1575}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{10 \cdot 25 + 3 \cdot 63}{1575}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $10$ by $25$ .

$\frac{250 + 3 \cdot 63}{1575}$

Step 2.6.2

Multiply $3$ by $63$ .

$\frac{250 + 189}{1575}$

Step 2.6.3

Add $250$ and $189$ .

$\frac{439}{1575}$