Linear Algebra Examples

[\begin{matrix} 6 & 8 2 & 5 \end{matrix}]

$[\begin{array}{c} 6 & 8 \\ 2 & 5 \end{array}]$

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

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

6 \cdot 5 - 2 \cdot 8

$6 \cdot 5 - 2 \cdot 8$

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

Multiply

6

$6$ by

5

$5$ .

30 - 2 \cdot 8

$30 - 2 \cdot 8$

Step 2.2.1.2

Multiply

- 2

$- 2$ by

8

$8$ .

30 - 16

$30 - 16$

30 - 16

$30 - 16$

Step 2.2.2

Subtract

16

$16$ from

30

$30$ .

14

$14$

14

$14$

14

$14$

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}{14} [\begin{matrix} 5 & - 8 - 2 & 6 \end{matrix}]

$\frac{1}{14} [\begin{array}{c} 5 & - 8 \\ - 2 & 6 \end{array}]$

Step 5

Multiply

\frac{1}{14}

$\frac{1}{14}$ by each element of the matrix.

[\begin{matrix} \frac{1}{14} \cdot 5 & \frac{1}{14} \cdot - 8 \frac{1}{14} \cdot - 2 & \frac{1}{14} \cdot 6 \end{matrix}]

$[\begin{array}{c} \frac{1}{14} \cdot 5 & \frac{1}{14} \cdot - 8 \\ \frac{1}{14} \cdot - 2 & \frac{1}{14} \cdot 6 \end{array}]$

Step 6

Simplify each element in the matrix.

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Step 6.1

Combine

\frac{1}{14}

$\frac{1}{14}$ and

5

$5$ .

[\begin{matrix} \frac{5}{14} & \frac{1}{14} \cdot - 8 \frac{1}{14} \cdot - 2 & \frac{1}{14} \cdot 6 \end{matrix}]

$[\begin{array}{c} \frac{5}{14} & \frac{1}{14} \cdot - 8 \\ \frac{1}{14} \cdot - 2 & \frac{1}{14} \cdot 6 \end{array}]$

Step 6.2

Cancel the common factor of

2

$2$ .

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Step 6.2.1

Factor

2

$2$ out of

14

$14$ .

⎡ ⎣ \begin{matrix} \frac{5}{14} & \frac{1}{2 (7)} \cdot - 8 \frac{1}{14} \cdot - 2 & \frac{1}{14} \cdot 6 \end{matrix} ⎤ ⎦

$[\begin{array}{c} \frac{5}{14} & \frac{1}{2 (7)} \cdot - 8 \\ \frac{1}{14} \cdot - 2 & \frac{1}{14} \cdot 6 \end{array}]$

Step 6.2.2

Factor

2

$2$ out of

- 8

$- 8$ .

[\begin{matrix} \frac{5}{14} & \frac{1}{2 \cdot 7} \cdot (2 \cdot - 4) \frac{1}{14} \cdot - 2 & \frac{1}{14} \cdot 6 \end{matrix}]

$[\begin{array}{c} \frac{5}{14} & \frac{1}{2 \cdot 7} \cdot (2 \cdot - 4) \\ \frac{1}{14} \cdot - 2 & \frac{1}{14} \cdot 6 \end{array}]$

Step 6.2.3

Cancel the common factor.

$[\begin{array}{c} \frac{5}{14} & \frac{1}{2 \cdot 7} \cdot (2 \cdot - 4) \\ \frac{1}{14} \cdot - 2 & \frac{1}{14} \cdot 6 \end{array}]$

Step 6.2.4

Rewrite the expression.

$[\begin{array}{c} \frac{5}{14} & \frac{1}{7} \cdot - 4 \\ \frac{1}{14} \cdot - 2 & \frac{1}{14} \cdot 6 \end{array}]$

Step 6.3

Combine

$\frac{1}{7}$ and

$- 4$ .

$[\begin{array}{c} \frac{5}{14} & \frac{- 4}{7} \\ \frac{1}{14} \cdot - 2 & \frac{1}{14} \cdot 6 \end{array}]$

Step 6.4

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{5}{14} & - \frac{4}{7} \\ \frac{1}{14} \cdot - 2 & \frac{1}{14} \cdot 6 \end{array}]$

Step 6.5

Cancel the common factor of

$2$ .

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Step 6.5.1

Factor

$2$ out of

$14$ .

$[\begin{array}{c} \frac{5}{14} & - \frac{4}{7} \\ \frac{1}{2 (7)} \cdot - 2 & \frac{1}{14} \cdot 6 \end{array}]$

Step 6.5.2

Factor

$2$ out of

$- 2$ .

$[\begin{array}{c} \frac{5}{14} & - \frac{4}{7} \\ \frac{1}{2 \cdot 7} \cdot (2 \cdot - 1) & \frac{1}{14} \cdot 6 \end{array}]$

Step 6.5.3

Cancel the common factor.

$[\begin{array}{c} \frac{5}{14} & - \frac{4}{7} \\ \frac{1}{2 \cdot 7} \cdot (2 \cdot - 1) & \frac{1}{14} \cdot 6 \end{array}]$

Step 6.5.4

Rewrite the expression.

$[\begin{array}{c} \frac{5}{14} & - \frac{4}{7} \\ \frac{1}{7} \cdot - 1 & \frac{1}{14} \cdot 6 \end{array}]$

Step 6.6

Combine

$\frac{1}{7}$ and

$- 1$ .

$[\begin{array}{c} \frac{5}{14} & - \frac{4}{7} \\ \frac{- 1}{7} & \frac{1}{14} \cdot 6 \end{array}]$

Step 6.7

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{5}{14} & - \frac{4}{7} \\ - \frac{1}{7} & \frac{1}{14} \cdot 6 \end{array}]$

Step 6.8

Cancel the common factor of

$2$ .

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Step 6.8.1

Factor

$2$ out of

$14$ .

$[\begin{array}{c} \frac{5}{14} & - \frac{4}{7} \\ - \frac{1}{7} & \frac{1}{2 (7)} \cdot 6 \end{array}]$

Step 6.8.2

Factor

$2$ out of

$6$ .

$[\begin{array}{c} \frac{5}{14} & - \frac{4}{7} \\ - \frac{1}{7} & \frac{1}{2 \cdot 7} \cdot (2 \cdot 3) \end{array}]$

Step 6.8.3

Cancel the common factor.

$[\begin{array}{c} \frac{5}{14} & - \frac{4}{7} \\ - \frac{1}{7} & \frac{1}{2 \cdot 7} \cdot (2 \cdot 3) \end{array}]$

Step 6.8.4

Rewrite the expression.

$[\begin{array}{c} \frac{5}{14} & - \frac{4}{7} \\ - \frac{1}{7} & \frac{1}{7} \cdot 3 \end{array}]$

Step 6.9

Combine

$\frac{1}{7}$ and

$3$ .

$[\begin{array}{c} \frac{5}{14} & - \frac{4}{7} \\ - \frac{1}{7} & \frac{3}{7} \end{array}]$

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