Linear Algebra Examples

$[\begin{array}{c} - \frac{1}{4} & - \frac{1}{6} \\ \frac{1}{3} & \frac{4}{3} \end{array}]$

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

$- \frac{1}{4} \cdot \frac{4}{3} - \frac{1}{3} (- \frac{1}{6})$

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

Cancel the common factor of $4$ .

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Step 2.2.1.1.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$\frac{- 1}{4} \cdot \frac{4}{3} - \frac{1}{3} (- \frac{1}{6})$

Step 2.2.1.1.2

Cancel the common factor.

$\frac{- 1}{4} \cdot \frac{4}{3} - \frac{1}{3} (- \frac{1}{6})$

Step 2.2.1.1.3

Rewrite the expression.

$- \frac{1}{3} - \frac{1}{3} (- \frac{1}{6})$

Step 2.2.1.2

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

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Step 2.2.1.2.1

Multiply $- 1$ by $- 1$ .

$- \frac{1}{3} + 1 (\frac{1}{3}) \frac{1}{6}$

Step 2.2.1.2.2

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

$- \frac{1}{3} + \frac{1}{3} \cdot \frac{1}{6}$

Step 2.2.1.2.3

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

$- \frac{1}{3} + \frac{1}{3 \cdot 6}$

Step 2.2.1.2.4

Multiply $3$ by $6$ .

$- \frac{1}{3} + \frac{1}{18}$

Step 2.2.2

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{1}{3} \cdot \frac{6}{6} + \frac{1}{18}$

Step 2.2.3

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.3.1

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

$- \frac{6}{3 \cdot 6} + \frac{1}{18}$

Step 2.2.3.2

Multiply $3$ by $6$ .

$- \frac{6}{18} + \frac{1}{18}$

Step 2.2.4

Combine the numerators over the common denominator.

$\frac{- 6 + 1}{18}$

Step 2.2.5

Add $- 6$ and $1$ .

$\frac{- 5}{18}$

Step 2.2.6

Move the negative in front of the fraction.

$- \frac{5}{18}$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- \frac{5}{18}} [\begin{array}{c} \frac{4}{3} & \frac{1}{6} \\ - \frac{1}{3} & - \frac{1}{4} \end{array}]$

Step 5

Cancel the common factor of $1$ and $- 1$ .

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Step 5.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{- \frac{5}{18}} [\begin{array}{c} \frac{4}{3} & \frac{1}{6} \\ - \frac{1}{3} & - \frac{1}{4} \end{array}]$

Step 5.2

Move the negative in front of the fraction.

$- \frac{1}{\frac{5}{18}} [\begin{array}{c} \frac{4}{3} & \frac{1}{6} \\ - \frac{1}{3} & - \frac{1}{4} \end{array}]$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{18}{5})) [\begin{array}{c} \frac{4}{3} & \frac{1}{6} \\ - \frac{1}{3} & - \frac{1}{4} \end{array}]$

Step 7

Multiply $\frac{18}{5}$ by $1$ .

$- \frac{18}{5} [\begin{array}{c} \frac{4}{3} & \frac{1}{6} \\ - \frac{1}{3} & - \frac{1}{4} \end{array}]$

Step 8

Multiply $- \frac{18}{5}$ by each element of the matrix.

$[\begin{array}{c} - \frac{18}{5} \cdot \frac{4}{3} & - \frac{18}{5} \cdot \frac{1}{6} \\ - \frac{18}{5} (- \frac{1}{3}) & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9

Simplify each element in the matrix.

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Step 9.1

Cancel the common factor of $3$ .

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Step 9.1.1

Move the leading negative in $- \frac{18}{5}$ into the numerator.

$[\begin{array}{c} \frac{- 18}{5} \cdot \frac{4}{3} & - \frac{18}{5} \cdot \frac{1}{6} \\ - \frac{18}{5} (- \frac{1}{3}) & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.1.2

Factor $3$ out of $- 18$ .

$[\begin{array}{c} \frac{3 (- 6)}{5} \cdot \frac{4}{3} & - \frac{18}{5} \cdot \frac{1}{6} \\ - \frac{18}{5} (- \frac{1}{3}) & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.1.3

Cancel the common factor.

$[\begin{array}{c} \frac{3 \cdot - 6}{5} \cdot \frac{4}{3} & - \frac{18}{5} \cdot \frac{1}{6} \\ - \frac{18}{5} (- \frac{1}{3}) & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.1.4

Rewrite the expression.

$[\begin{array}{c} \frac{- 6}{5} \cdot 4 & - \frac{18}{5} \cdot \frac{1}{6} \\ - \frac{18}{5} (- \frac{1}{3}) & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.2

Combine $\frac{- 6}{5}$ and $4$ .

$[\begin{array}{c} \frac{- 6 \cdot 4}{5} & - \frac{18}{5} \cdot \frac{1}{6} \\ - \frac{18}{5} (- \frac{1}{3}) & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.3

Multiply $- 6$ by $4$ .

$[\begin{array}{c} \frac{- 24}{5} & - \frac{18}{5} \cdot \frac{1}{6} \\ - \frac{18}{5} (- \frac{1}{3}) & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.4

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{24}{5} & - \frac{18}{5} \cdot \frac{1}{6} \\ - \frac{18}{5} (- \frac{1}{3}) & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.5

Cancel the common factor of $6$ .

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Step 9.5.1

Move the leading negative in $- \frac{18}{5}$ into the numerator.

$[\begin{array}{c} - \frac{24}{5} & \frac{- 18}{5} \cdot \frac{1}{6} \\ - \frac{18}{5} (- \frac{1}{3}) & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.5.2

Factor $6$ out of $- 18$ .

$[\begin{array}{c} - \frac{24}{5} & \frac{6 (- 3)}{5} \cdot \frac{1}{6} \\ - \frac{18}{5} (- \frac{1}{3}) & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.5.3

Cancel the common factor.

$[\begin{array}{c} - \frac{24}{5} & \frac{6 \cdot - 3}{5} \cdot \frac{1}{6} \\ - \frac{18}{5} (- \frac{1}{3}) & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.5.4

Rewrite the expression.

$[\begin{array}{c} - \frac{24}{5} & \frac{- 3}{5} \\ - \frac{18}{5} (- \frac{1}{3}) & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.6

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ - \frac{18}{5} (- \frac{1}{3}) & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.7

Cancel the common factor of $3$ .

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Step 9.7.1

Move the leading negative in $- \frac{18}{5}$ into the numerator.

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ \frac{- 18}{5} (- \frac{1}{3}) & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.7.2

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

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ \frac{- 18}{5} \cdot \frac{- 1}{3} & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.7.3

Factor $3$ out of $- 18$ .

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ \frac{3 (- 6)}{5} \cdot \frac{- 1}{3} & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.7.4

Cancel the common factor.

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ \frac{3 \cdot - 6}{5} \cdot \frac{- 1}{3} & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.7.5

Rewrite the expression.

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ \frac{- 6}{5} \cdot - 1 & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.8

Combine $\frac{- 6}{5}$ and $- 1$ .

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ \frac{- 6 \cdot - 1}{5} & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.9

Multiply $- 6$ by $- 1$ .

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ \frac{6}{5} & - \frac{18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.10

Cancel the common factor of $2$ .

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Step 9.10.1

Move the leading negative in $- \frac{18}{5}$ into the numerator.

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ \frac{6}{5} & \frac{- 18}{5} (- \frac{1}{4}) \end{array}]$

Step 9.10.2

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ \frac{6}{5} & \frac{- 18}{5} \cdot \frac{- 1}{4} \end{array}]$

Step 9.10.3

Factor $2$ out of $- 18$ .

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ \frac{6}{5} & \frac{2 (- 9)}{5} \cdot \frac{- 1}{4} \end{array}]$

Step 9.10.4

Factor $2$ out of $4$ .

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ \frac{6}{5} & \frac{2 \cdot - 9}{5} \cdot \frac{- 1}{2 \cdot 2} \end{array}]$

Step 9.10.5

Cancel the common factor.

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ \frac{6}{5} & \frac{2 \cdot - 9}{5} \cdot \frac{- 1}{2 \cdot 2} \end{array}]$

Step 9.10.6

Rewrite the expression.

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ \frac{6}{5} & \frac{- 9}{5} \cdot \frac{- 1}{2} \end{array}]$

Step 9.11

Multiply $\frac{- 9}{5}$ by $\frac{- 1}{2}$ .

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ \frac{6}{5} & \frac{- 9 \cdot - 1}{5 \cdot 2} \end{array}]$

Step 9.12

Multiply $- 9$ by $- 1$ .

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ \frac{6}{5} & \frac{9}{5 \cdot 2} \end{array}]$

Step 9.13

Multiply $5$ by $2$ .

$[\begin{array}{c} - \frac{24}{5} & - \frac{3}{5} \\ \frac{6}{5} & \frac{9}{10} \end{array}]$