Linear Algebra Examples

$[\begin{array}{c} 7 & 8 \\ \frac{2}{3} & \frac{1}{3} \end{array}]$

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

$7 (\frac{1}{3}) - \frac{2}{3} \cdot 8$

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

Combine $7$ and $\frac{1}{3}$ .

$\frac{7}{3} - \frac{2}{3} \cdot 8$

Step 2.2.1.2

Multiply $- \frac{2}{3} \cdot 8$ .

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Step 2.2.1.2.1

Multiply $8$ by $- 1$ .

$\frac{7}{3} - 8 (\frac{2}{3})$

Step 2.2.1.2.2

Combine $- 8$ and $\frac{2}{3}$ .

$\frac{7}{3} + \frac{- 8 \cdot 2}{3}$

Step 2.2.1.2.3

Multiply $- 8$ by $2$ .

$\frac{7}{3} + \frac{- 16}{3}$

Step 2.2.1.3

Move the negative in front of the fraction.

$\frac{7}{3} - \frac{16}{3}$

Step 2.2.2

Combine the numerators over the common denominator.

$\frac{7 - 16}{3}$

Step 2.2.3

Subtract $16$ from $7$ .

$\frac{- 9}{3}$

Step 2.2.4

Divide $- 9$ by $3$ .

$- 3$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- 3} [\begin{array}{c} \frac{1}{3} & - 8 \\ - \frac{2}{3} & 7 \end{array}]$

Step 5

Move the negative in front of the fraction.

$- \frac{1}{3} [\begin{array}{c} \frac{1}{3} & - 8 \\ - \frac{2}{3} & 7 \end{array}]$

Step 6

Multiply $- \frac{1}{3}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{3} \cdot \frac{1}{3} & - \frac{1}{3} \cdot - 8 \\ - \frac{1}{3} (- \frac{2}{3}) & - \frac{1}{3} \cdot 7 \end{array}]$

Step 7

Simplify each element in the matrix.

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Step 7.1

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

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Step 7.1.1

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

$[\begin{array}{c} - \frac{1}{3 \cdot 3} & - \frac{1}{3} \cdot - 8 \\ - \frac{1}{3} (- \frac{2}{3}) & - \frac{1}{3} \cdot 7 \end{array}]$

Step 7.1.2

Multiply $3$ by $3$ .

$[\begin{array}{c} - \frac{1}{9} & - \frac{1}{3} \cdot - 8 \\ - \frac{1}{3} (- \frac{2}{3}) & - \frac{1}{3} \cdot 7 \end{array}]$

Step 7.2

Multiply $- \frac{1}{3} \cdot - 8$ .

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Step 7.2.1

Multiply $- 8$ by $- 1$ .

$[\begin{array}{c} - \frac{1}{9} & 8 (\frac{1}{3}) \\ - \frac{1}{3} (- \frac{2}{3}) & - \frac{1}{3} \cdot 7 \end{array}]$

Step 7.2.2

Combine $8$ and $\frac{1}{3}$ .

$[\begin{array}{c} - \frac{1}{9} & \frac{8}{3} \\ - \frac{1}{3} (- \frac{2}{3}) & - \frac{1}{3} \cdot 7 \end{array}]$

Step 7.3

Multiply $- \frac{1}{3} (- \frac{2}{3})$ .

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Step 7.3.1

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} - \frac{1}{9} & \frac{8}{3} \\ 1 (\frac{1}{3}) \frac{2}{3} & - \frac{1}{3} \cdot 7 \end{array}]$

Step 7.3.2

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

$[\begin{array}{c} - \frac{1}{9} & \frac{8}{3} \\ \frac{1}{3} \cdot \frac{2}{3} & - \frac{1}{3} \cdot 7 \end{array}]$

Step 7.3.3

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

$[\begin{array}{c} - \frac{1}{9} & \frac{8}{3} \\ \frac{2}{3 \cdot 3} & - \frac{1}{3} \cdot 7 \end{array}]$

Step 7.3.4

Multiply $3$ by $3$ .

$[\begin{array}{c} - \frac{1}{9} & \frac{8}{3} \\ \frac{2}{9} & - \frac{1}{3} \cdot 7 \end{array}]$

Step 7.4

Multiply $- \frac{1}{3} \cdot 7$ .

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Step 7.4.1

Multiply $7$ by $- 1$ .

$[\begin{array}{c} - \frac{1}{9} & \frac{8}{3} \\ \frac{2}{9} & - 7 (\frac{1}{3}) \end{array}]$

Step 7.4.2

Combine $- 7$ and $\frac{1}{3}$ .

$[\begin{array}{c} - \frac{1}{9} & \frac{8}{3} \\ \frac{2}{9} & \frac{- 7}{3} \end{array}]$

Step 7.5

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{9} & \frac{8}{3} \\ \frac{2}{9} & - \frac{7}{3} \end{array}]$