Linear Algebra Examples

$[\begin{array}{c} 5 & 10 \\ 10 & 30 \end{array}]$

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

$5 \cdot 30 - 10 \cdot 10$

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 $5$ by $30$ .

$150 - 10 \cdot 10$

Step 2.2.1.2

Multiply $- 10$ by $10$ .

$150 - 100$

Step 2.2.2

Subtract $100$ from $150$ .

$50$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{50} [\begin{array}{c} 30 & - 10 \\ - 10 & 5 \end{array}]$

Step 5

Multiply $\frac{1}{50}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{50} \cdot 30 & \frac{1}{50} \cdot - 10 \\ \frac{1}{50} \cdot - 10 & \frac{1}{50} \cdot 5 \end{array}]$

Step 6

Simplify each element in the matrix.

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Step 6.1

Cancel the common factor of $10$ .

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Step 6.1.1

Factor $10$ out of $50$ .

$[\begin{array}{c} \frac{1}{10 (5)} \cdot 30 & \frac{1}{50} \cdot - 10 \\ \frac{1}{50} \cdot - 10 & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.1.2

Factor $10$ out of $30$ .

$[\begin{array}{c} \frac{1}{10 \cdot 5} \cdot (10 \cdot 3) & \frac{1}{50} \cdot - 10 \\ \frac{1}{50} \cdot - 10 & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.1.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{10 \cdot 5} \cdot (10 \cdot 3) & \frac{1}{50} \cdot - 10 \\ \frac{1}{50} \cdot - 10 & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.1.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{5} \cdot 3 & \frac{1}{50} \cdot - 10 \\ \frac{1}{50} \cdot - 10 & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.2

Combine $\frac{1}{5}$ and $3$ .

$[\begin{array}{c} \frac{3}{5} & \frac{1}{50} \cdot - 10 \\ \frac{1}{50} \cdot - 10 & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.3

Cancel the common factor of $10$ .

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Step 6.3.1

Factor $10$ out of $50$ .

$[\begin{array}{c} \frac{3}{5} & \frac{1}{10 (5)} \cdot - 10 \\ \frac{1}{50} \cdot - 10 & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.3.2

Factor $10$ out of $- 10$ .

$[\begin{array}{c} \frac{3}{5} & \frac{1}{10 \cdot 5} \cdot (10 \cdot - 1) \\ \frac{1}{50} \cdot - 10 & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.3.3

Cancel the common factor.

$[\begin{array}{c} \frac{3}{5} & \frac{1}{10 \cdot 5} \cdot (10 \cdot - 1) \\ \frac{1}{50} \cdot - 10 & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.3.4

Rewrite the expression.

$[\begin{array}{c} \frac{3}{5} & \frac{1}{5} \cdot - 1 \\ \frac{1}{50} \cdot - 10 & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.4

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

$[\begin{array}{c} \frac{3}{5} & \frac{- 1}{5} \\ \frac{1}{50} \cdot - 10 & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.5

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{3}{5} & - \frac{1}{5} \\ \frac{1}{50} \cdot - 10 & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.6

Cancel the common factor of $10$ .

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Step 6.6.1

Factor $10$ out of $50$ .

$[\begin{array}{c} \frac{3}{5} & - \frac{1}{5} \\ \frac{1}{10 (5)} \cdot - 10 & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.6.2

Factor $10$ out of $- 10$ .

$[\begin{array}{c} \frac{3}{5} & - \frac{1}{5} \\ \frac{1}{10 \cdot 5} \cdot (10 \cdot - 1) & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.6.3

Cancel the common factor.

$[\begin{array}{c} \frac{3}{5} & - \frac{1}{5} \\ \frac{1}{10 \cdot 5} \cdot (10 \cdot - 1) & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.6.4

Rewrite the expression.

$[\begin{array}{c} \frac{3}{5} & - \frac{1}{5} \\ \frac{1}{5} \cdot - 1 & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.7

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

$[\begin{array}{c} \frac{3}{5} & - \frac{1}{5} \\ \frac{- 1}{5} & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.8

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{3}{5} & - \frac{1}{5} \\ - \frac{1}{5} & \frac{1}{50} \cdot 5 \end{array}]$

Step 6.9

Cancel the common factor of $5$ .

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Step 6.9.1

Factor $5$ out of $50$ .

$[\begin{array}{c} \frac{3}{5} & - \frac{1}{5} \\ - \frac{1}{5} & \frac{1}{5 (10)} \cdot 5 \end{array}]$

Step 6.9.2

Cancel the common factor.

$[\begin{array}{c} \frac{3}{5} & - \frac{1}{5} \\ - \frac{1}{5} & \frac{1}{5 \cdot 10} \cdot 5 \end{array}]$

Step 6.9.3

Rewrite the expression.

$[\begin{array}{c} \frac{3}{5} & - \frac{1}{5} \\ - \frac{1}{5} & \frac{1}{10} \end{array}]$