Linear Algebra Examples

[\begin{matrix} 7 & 8 \frac{2}{3} & \frac{1}{3} \end{matrix}]

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

Step 1

The inverse of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

\frac{1}{a d - b c} [\begin{matrix} d & - b - c & a \end{matrix}]

$\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where

a d - b c

$a d - b c$ is the determinant.

Step 2

Find the determinant.

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Step 2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

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

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

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

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

$7$ and

\frac{1}{3}

$\frac{1}{3}$ .

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

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

Step 2.2.1.2

Multiply

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

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

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Step 2.2.1.2.1

Multiply

8

$8$ by

- 1

$- 1$ .

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

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

Step 2.2.1.2.2

Combine

- 8

$- 8$ and

\frac{2}{3}

$\frac{2}{3}$ .

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

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

Step 2.2.1.2.3

Multiply

- 8

$- 8$ by

2

$2$ .

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

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

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

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

Step 2.2.1.3

Move the negative in front of the fraction.

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

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

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

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

Step 2.2.2

Combine the numerators over the common denominator.

\frac{7 - 16}{3}

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

Step 2.2.3

Subtract

16

$16$ from

7

$7$ .

\frac{- 9}{3}

$\frac{- 9}{3}$

Step 2.2.4

Divide

- 9

$- 9$ by

3

$3$ .

- 3

$- 3$

- 3

$- 3$

- 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{matrix} \frac{1}{3} & - 8 - \frac{2}{3} & 7 \end{matrix}]

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

Step 5

Move the negative in front of the fraction.

- \frac{1}{3} [\begin{matrix} \frac{1}{3} & - 8 - \frac{2}{3} & 7 \end{matrix}]

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

Step 6

Multiply

- \frac{1}{3}

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

⎡ ⎢ ⎣ \begin{matrix} - \frac{1}{3} \cdot \frac{1}{3} & - \frac{1}{3} \cdot - 8 - \frac{1}{3} (- \frac{2}{3}) & - \frac{1}{3} \cdot 7 \end{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}

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

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Step 7.1.1

Multiply

\frac{1}{3}

$\frac{1}{3}$ by

\frac{1}{3}

$\frac{1}{3}$ .

⎡ ⎢ ⎣ \begin{matrix} - \frac{1}{3 \cdot 3} & - \frac{1}{3} \cdot - 8 - \frac{1}{3} (- \frac{2}{3}) & - \frac{1}{3} \cdot 7 \end{matrix} ⎤ ⎥ ⎦

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

Step 7.1.2

Multiply

3

$3$ by

3

$3$ .

⎡ ⎢ ⎣ \begin{matrix} - \frac{1}{9} & - \frac{1}{3} \cdot - 8 - \frac{1}{3} (- \frac{2}{3}) & - \frac{1}{3} \cdot 7 \end{matrix} ⎤ ⎥ ⎦

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

⎡ ⎢ ⎣ \begin{matrix} - \frac{1}{9} & - \frac{1}{3} \cdot - 8 - \frac{1}{3} (- \frac{2}{3}) & - \frac{1}{3} \cdot 7 \end{matrix} ⎤ ⎥ ⎦

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

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.

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