Linear Algebra Examples

$[\begin{array}{c} 2 & - 8 \\ 11 & 11 \end{array}]$

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

$2 \cdot 11 - 11 \cdot - 8$

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 $2$ by $11$ .

$22 - 11 \cdot - 8$

Step 2.2.1.2

Multiply $- 11$ by $- 8$ .

$22 + 88$

Step 2.2.2

Add $22$ and $88$ .

$110$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{110} [\begin{array}{c} 11 & 8 \\ - 11 & 2 \end{array}]$

Step 5

Multiply $\frac{1}{110}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{110} \cdot 11 & \frac{1}{110} \cdot 8 \\ \frac{1}{110} \cdot - 11 & \frac{1}{110} \cdot 2 \end{array}]$

Step 6

Simplify each element in the matrix.

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Step 6.1

Cancel the common factor of $11$ .

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Step 6.1.1

Factor $11$ out of $110$ .

$[\begin{array}{c} \frac{1}{11 (10)} \cdot 11 & \frac{1}{110} \cdot 8 \\ \frac{1}{110} \cdot - 11 & \frac{1}{110} \cdot 2 \end{array}]$

Step 6.1.2

Cancel the common factor.

$[\begin{array}{c} \frac{1}{11 \cdot 10} \cdot 11 & \frac{1}{110} \cdot 8 \\ \frac{1}{110} \cdot - 11 & \frac{1}{110} \cdot 2 \end{array}]$

Step 6.1.3

Rewrite the expression.

$[\begin{array}{c} \frac{1}{10} & \frac{1}{110} \cdot 8 \\ \frac{1}{110} \cdot - 11 & \frac{1}{110} \cdot 2 \end{array}]$

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Factor $2$ out of $110$ .

$[\begin{array}{c} \frac{1}{10} & \frac{1}{2 (55)} \cdot 8 \\ \frac{1}{110} \cdot - 11 & \frac{1}{110} \cdot 2 \end{array}]$

Step 6.2.2

Factor $2$ out of $8$ .

$[\begin{array}{c} \frac{1}{10} & \frac{1}{2 \cdot 55} \cdot (2 \cdot 4) \\ \frac{1}{110} \cdot - 11 & \frac{1}{110} \cdot 2 \end{array}]$

Step 6.2.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{10} & \frac{1}{2 \cdot 55} \cdot (2 \cdot 4) \\ \frac{1}{110} \cdot - 11 & \frac{1}{110} \cdot 2 \end{array}]$

Step 6.2.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{10} & \frac{1}{55} \cdot 4 \\ \frac{1}{110} \cdot - 11 & \frac{1}{110} \cdot 2 \end{array}]$

Step 6.3

Combine $\frac{1}{55}$ and $4$ .

$[\begin{array}{c} \frac{1}{10} & \frac{4}{55} \\ \frac{1}{110} \cdot - 11 & \frac{1}{110} \cdot 2 \end{array}]$

Step 6.4

Cancel the common factor of $11$ .

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Step 6.4.1

Factor $11$ out of $110$ .

$[\begin{array}{c} \frac{1}{10} & \frac{4}{55} \\ \frac{1}{11 (10)} \cdot - 11 & \frac{1}{110} \cdot 2 \end{array}]$

Step 6.4.2

Factor $11$ out of $- 11$ .

$[\begin{array}{c} \frac{1}{10} & \frac{4}{55} \\ \frac{1}{11 \cdot 10} \cdot (11 \cdot - 1) & \frac{1}{110} \cdot 2 \end{array}]$

Step 6.4.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{10} & \frac{4}{55} \\ \frac{1}{11 \cdot 10} \cdot (11 \cdot - 1) & \frac{1}{110} \cdot 2 \end{array}]$

Step 6.4.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{10} & \frac{4}{55} \\ \frac{1}{10} \cdot - 1 & \frac{1}{110} \cdot 2 \end{array}]$

Step 6.5

Combine $\frac{1}{10}$ and $- 1$ .

$[\begin{array}{c} \frac{1}{10} & \frac{4}{55} \\ \frac{- 1}{10} & \frac{1}{110} \cdot 2 \end{array}]$

Step 6.6

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{10} & \frac{4}{55} \\ - \frac{1}{10} & \frac{1}{110} \cdot 2 \end{array}]$

Step 6.7

Cancel the common factor of $2$ .

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Step 6.7.1

Factor $2$ out of $110$ .

$[\begin{array}{c} \frac{1}{10} & \frac{4}{55} \\ - \frac{1}{10} & \frac{1}{2 (55)} \cdot 2 \end{array}]$

Step 6.7.2

Cancel the common factor.

$[\begin{array}{c} \frac{1}{10} & \frac{4}{55} \\ - \frac{1}{10} & \frac{1}{2 \cdot 55} \cdot 2 \end{array}]$

Step 6.7.3

Rewrite the expression.

$[\begin{array}{c} \frac{1}{10} & \frac{4}{55} \\ - \frac{1}{10} & \frac{1}{55} \end{array}]$