Linear Algebra Examples

$[\begin{array}{c} 8 & 9 \\ 2 & 0 \end{array}]$

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

$8 \cdot 0 - 2 \cdot 9$

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 $8$ by $0$ .

$0 - 2 \cdot 9$

Step 2.2.1.2

Multiply $- 2$ by $9$ .

$0 - 18$

Step 2.2.2

Subtract $18$ from $0$ .

$- 18$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- 18} [\begin{array}{c} 0 & - 9 \\ - 2 & 8 \end{array}]$

Step 5

Move the negative in front of the fraction.

$- \frac{1}{18} [\begin{array}{c} 0 & - 9 \\ - 2 & 8 \end{array}]$

Step 6

Multiply $- \frac{1}{18}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{18} \cdot 0 & - \frac{1}{18} \cdot - 9 \\ - \frac{1}{18} \cdot - 2 & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7

Simplify each element in the matrix.

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Step 7.1

Multiply $- \frac{1}{18} \cdot 0$ .

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Step 7.1.1

Multiply $0$ by $- 1$ .

$[\begin{array}{c} 0 (\frac{1}{18}) & - \frac{1}{18} \cdot - 9 \\ - \frac{1}{18} \cdot - 2 & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7.1.2

Multiply $0$ by $\frac{1}{18}$ .

$[\begin{array}{c} 0 & - \frac{1}{18} \cdot - 9 \\ - \frac{1}{18} \cdot - 2 & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7.2

Cancel the common factor of $9$ .

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Step 7.2.1

Move the leading negative in $- \frac{1}{18}$ into the numerator.

$[\begin{array}{c} 0 & \frac{- 1}{18} \cdot - 9 \\ - \frac{1}{18} \cdot - 2 & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7.2.2

Factor $9$ out of $18$ .

$[\begin{array}{c} 0 & \frac{- 1}{9 (2)} \cdot - 9 \\ - \frac{1}{18} \cdot - 2 & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7.2.3

Factor $9$ out of $- 9$ .

$[\begin{array}{c} 0 & \frac{- 1}{9 \cdot 2} \cdot (9 \cdot - 1) \\ - \frac{1}{18} \cdot - 2 & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7.2.4

Cancel the common factor.

$[\begin{array}{c} 0 & \frac{- 1}{9 \cdot 2} \cdot (9 \cdot - 1) \\ - \frac{1}{18} \cdot - 2 & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7.2.5

Rewrite the expression.

$[\begin{array}{c} 0 & \frac{- 1}{2} \cdot - 1 \\ - \frac{1}{18} \cdot - 2 & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7.3

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

$[\begin{array}{c} 0 & \frac{- - 1}{2} \\ - \frac{1}{18} \cdot - 2 & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7.4

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} 0 & \frac{1}{2} \\ - \frac{1}{18} \cdot - 2 & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7.5

Cancel the common factor of $2$ .

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Step 7.5.1

Move the leading negative in $- \frac{1}{18}$ into the numerator.

$[\begin{array}{c} 0 & \frac{1}{2} \\ \frac{- 1}{18} \cdot - 2 & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7.5.2

Factor $2$ out of $18$ .

$[\begin{array}{c} 0 & \frac{1}{2} \\ \frac{- 1}{2 (9)} \cdot - 2 & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7.5.3

Factor $2$ out of $- 2$ .

$[\begin{array}{c} 0 & \frac{1}{2} \\ \frac{- 1}{2 \cdot 9} \cdot (2 \cdot - 1) & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7.5.4

Cancel the common factor.

$[\begin{array}{c} 0 & \frac{1}{2} \\ \frac{- 1}{2 \cdot 9} \cdot (2 \cdot - 1) & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7.5.5

Rewrite the expression.

$[\begin{array}{c} 0 & \frac{1}{2} \\ \frac{- 1}{9} \cdot - 1 & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7.6

Combine $\frac{- 1}{9}$ and $- 1$ .

$[\begin{array}{c} 0 & \frac{1}{2} \\ \frac{- - 1}{9} & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7.7

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} 0 & \frac{1}{2} \\ \frac{1}{9} & - \frac{1}{18} \cdot 8 \end{array}]$

Step 7.8

Cancel the common factor of $2$ .

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Step 7.8.1

Move the leading negative in $- \frac{1}{18}$ into the numerator.

$[\begin{array}{c} 0 & \frac{1}{2} \\ \frac{1}{9} & \frac{- 1}{18} \cdot 8 \end{array}]$

Step 7.8.2

Factor $2$ out of $18$ .

$[\begin{array}{c} 0 & \frac{1}{2} \\ \frac{1}{9} & \frac{- 1}{2 (9)} \cdot 8 \end{array}]$

Step 7.8.3

Factor $2$ out of $8$ .

$[\begin{array}{c} 0 & \frac{1}{2} \\ \frac{1}{9} & \frac{- 1}{2 \cdot 9} \cdot (2 \cdot 4) \end{array}]$

Step 7.8.4

Cancel the common factor.

$[\begin{array}{c} 0 & \frac{1}{2} \\ \frac{1}{9} & \frac{- 1}{2 \cdot 9} \cdot (2 \cdot 4) \end{array}]$

Step 7.8.5

Rewrite the expression.

$[\begin{array}{c} 0 & \frac{1}{2} \\ \frac{1}{9} & \frac{- 1}{9} \cdot 4 \end{array}]$

Step 7.9

Combine $\frac{- 1}{9}$ and $4$ .

$[\begin{array}{c} 0 & \frac{1}{2} \\ \frac{1}{9} & \frac{- 1 \cdot 4}{9} \end{array}]$

Step 7.10

Multiply $- 1$ by $4$ .

$[\begin{array}{c} 0 & \frac{1}{2} \\ \frac{1}{9} & \frac{- 4}{9} \end{array}]$

Step 7.11

Move the negative in front of the fraction.

$[\begin{array}{c} 0 & \frac{1}{2} \\ \frac{1}{9} & - \frac{4}{9} \end{array}]$

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