Algebra Examples

$[\begin{array}{c} \frac{1}{3} & \frac{1}{3} \\ \frac{1}{5} & \frac{2}{5} \end{array}]$

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

$\frac{1}{3} \cdot \frac{2}{5} - \frac{1}{5} \cdot \frac{1}{3}$

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 $\frac{1}{3} \cdot \frac{2}{5}$ .

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Step 2.2.1.1.1

Multiply $\frac{1}{3}$ by $\frac{2}{5}$ .

$\frac{2}{3 \cdot 5} - \frac{1}{5} \cdot \frac{1}{3}$

Step 2.2.1.1.2

Multiply $3$ by $5$ .

$\frac{2}{15} - \frac{1}{5} \cdot \frac{1}{3}$

Step 2.2.1.2

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

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Step 2.2.1.2.1

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

$\frac{2}{15} - \frac{1}{3 \cdot 5}$

Step 2.2.1.2.2

Multiply $3$ by $5$ .

$\frac{2}{15} - \frac{1}{15}$

Step 2.2.2

Combine the numerators over the common denominator.

$\frac{2 - 1}{15}$

Step 2.2.3

Subtract $1$ from $2$ .

$\frac{1}{15}$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{\frac{1}{15}} [\begin{array}{c} \frac{2}{5} & - \frac{1}{3} \\ - \frac{1}{5} & \frac{1}{3} \end{array}]$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 15 [\begin{array}{c} \frac{2}{5} & - \frac{1}{3} \\ - \frac{1}{5} & \frac{1}{3} \end{array}]$

Step 6

Multiply $15$ by $1$ .

$15 [\begin{array}{c} \frac{2}{5} & - \frac{1}{3} \\ - \frac{1}{5} & \frac{1}{3} \end{array}]$

Step 7

Multiply $15$ by each element of the matrix.

$[\begin{array}{c} 15 (\frac{2}{5}) & 15 (- \frac{1}{3}) \\ 15 (- \frac{1}{5}) & 15 (\frac{1}{3}) \end{array}]$

Step 8

Simplify each element in the matrix.

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Step 8.1

Cancel the common factor of $5$ .

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Step 8.1.1

Factor $5$ out of $15$ .

$[\begin{array}{c} 5 (3) \frac{2}{5} & 15 (- \frac{1}{3}) \\ 15 (- \frac{1}{5}) & 15 (\frac{1}{3}) \end{array}]$

Step 8.1.2

Cancel the common factor.

$[\begin{array}{c} 5 \cdot 3 \frac{2}{5} & 15 (- \frac{1}{3}) \\ 15 (- \frac{1}{5}) & 15 (\frac{1}{3}) \end{array}]$

Step 8.1.3

Rewrite the expression.

$[\begin{array}{c} 3 \cdot 2 & 15 (- \frac{1}{3}) \\ 15 (- \frac{1}{5}) & 15 (\frac{1}{3}) \end{array}]$

Step 8.2

Multiply $3$ by $2$ .

$[\begin{array}{c} 6 & 15 (- \frac{1}{3}) \\ 15 (- \frac{1}{5}) & 15 (\frac{1}{3}) \end{array}]$

Step 8.3

Cancel the common factor of $3$ .

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Step 8.3.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$[\begin{array}{c} 6 & 15 (\frac{- 1}{3}) \\ 15 (- \frac{1}{5}) & 15 (\frac{1}{3}) \end{array}]$

Step 8.3.2

Factor $3$ out of $15$ .

$[\begin{array}{c} 6 & 3 (5) \frac{- 1}{3} \\ 15 (- \frac{1}{5}) & 15 (\frac{1}{3}) \end{array}]$

Step 8.3.3

Cancel the common factor.

$[\begin{array}{c} 6 & 3 \cdot 5 \frac{- 1}{3} \\ 15 (- \frac{1}{5}) & 15 (\frac{1}{3}) \end{array}]$

Step 8.3.4

Rewrite the expression.

$[\begin{array}{c} 6 & 5 \cdot - 1 \\ 15 (- \frac{1}{5}) & 15 (\frac{1}{3}) \end{array}]$

Step 8.4

Multiply $5$ by $- 1$ .

$[\begin{array}{c} 6 & - 5 \\ 15 (- \frac{1}{5}) & 15 (\frac{1}{3}) \end{array}]$

Step 8.5

Cancel the common factor of $5$ .

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Step 8.5.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$[\begin{array}{c} 6 & - 5 \\ 15 (\frac{- 1}{5}) & 15 (\frac{1}{3}) \end{array}]$

Step 8.5.2

Factor $5$ out of $15$ .

$[\begin{array}{c} 6 & - 5 \\ 5 (3) \frac{- 1}{5} & 15 (\frac{1}{3}) \end{array}]$

Step 8.5.3

Cancel the common factor.

$[\begin{array}{c} 6 & - 5 \\ 5 \cdot 3 \frac{- 1}{5} & 15 (\frac{1}{3}) \end{array}]$

Step 8.5.4

Rewrite the expression.

$[\begin{array}{c} 6 & - 5 \\ 3 \cdot - 1 & 15 (\frac{1}{3}) \end{array}]$

Step 8.6

Multiply $3$ by $- 1$ .

$[\begin{array}{c} 6 & - 5 \\ - 3 & 15 (\frac{1}{3}) \end{array}]$

Step 8.7

Cancel the common factor of $3$ .

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Step 8.7.1

Factor $3$ out of $15$ .

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

Step 8.7.2

Cancel the common factor.

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

Step 8.7.3

Rewrite the expression.

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