Linear Algebra Examples

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

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

$\frac{1}{2} \cdot 1 - 1 \cdot 1$

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

$\frac{1}{2} - 1 \cdot 1$

Step 2.2.1.2

Multiply $- 1$ by $1$ .

$\frac{1}{2} - 1$

Step 2.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} - 1 \cdot \frac{2}{2}$

Step 2.2.3

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{2} + \frac{- 1 \cdot 2}{2}$

Step 2.2.4

Combine the numerators over the common denominator.

$\frac{1 - 1 \cdot 2}{2}$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $- 1$ by $2$ .

$\frac{1 - 2}{2}$

Step 2.2.5.2

Subtract $2$ from $1$ .

$\frac{- 1}{2}$

Step 2.2.6

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

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

Step 5

Cancel the common factor of $1$ and $- 1$ .

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Step 5.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 5.2

Move the negative in front of the fraction.

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

Step 6

Multiply the numerator by the reciprocal of the denominator.

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

Step 7

Multiply $- (1 \cdot 2)$ .

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Step 7.1

Multiply $2$ by $1$ .

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

Step 7.2

Multiply $- 1$ by $2$ .

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

Step 8

Multiply $- 2$ by each element of the matrix.

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

Step 9

Simplify each element in the matrix.

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Step 9.1

Multiply $- 2$ by $1$ .

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

Step 9.2

Multiply $- 2$ by $- 1$ .

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

Step 9.3

Multiply $- 2$ by $- 1$ .

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

Step 9.4

Cancel the common factor of $2$ .

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Step 9.4.1

Factor $2$ out of $- 2$ .

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

Step 9.4.2

Cancel the common factor.

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

Step 9.4.3

Rewrite the expression.

$[\begin{array}{c} - 2 & 2 \\ 2 & - 1 \end{array}]$