Linear Algebra Examples

$[\begin{array}{c} 22000 & 45000 \\ 18000 & 27000 \end{array}]$

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

$22000 \cdot 27000 - 18000 \cdot 45000$

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 $22000$ by $27000$ .

$594000000 - 18000 \cdot 45000$

Step 2.2.1.2

Multiply $- 18000$ by $45000$ .

$594000000 - 810000000$

Step 2.2.2

Subtract $810000000$ from $594000000$ .

$- 216000000$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- 216000000} [\begin{array}{c} 27000 & - 45000 \\ - 18000 & 22000 \end{array}]$

Step 5

Move the negative in front of the fraction.

$- \frac{1}{216000000} [\begin{array}{c} 27000 & - 45000 \\ - 18000 & 22000 \end{array}]$

Step 6

Multiply $- \frac{1}{216000000}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{216000000} \cdot 27000 & - \frac{1}{216000000} \cdot - 45000 \\ - \frac{1}{216000000} \cdot - 18000 & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7

Simplify each element in the matrix.

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Step 7.1

Cancel the common factor of $27000$ .

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Step 7.1.1

Move the leading negative in $- \frac{1}{216000000}$ into the numerator.

$[\begin{array}{c} \frac{- 1}{216000000} \cdot 27000 & - \frac{1}{216000000} \cdot - 45000 \\ - \frac{1}{216000000} \cdot - 18000 & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.1.2

Factor $27000$ out of $216000000$ .

$[\begin{array}{c} \frac{- 1}{27000 (8000)} \cdot 27000 & - \frac{1}{216000000} \cdot - 45000 \\ - \frac{1}{216000000} \cdot - 18000 & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.1.3

Cancel the common factor.

$[\begin{array}{c} \frac{- 1}{27000 \cdot 8000} \cdot 27000 & - \frac{1}{216000000} \cdot - 45000 \\ - \frac{1}{216000000} \cdot - 18000 & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.1.4

Rewrite the expression.

$[\begin{array}{c} \frac{- 1}{8000} & - \frac{1}{216000000} \cdot - 45000 \\ - \frac{1}{216000000} \cdot - 18000 & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.2

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{8000} & - \frac{1}{216000000} \cdot - 45000 \\ - \frac{1}{216000000} \cdot - 18000 & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.3

Cancel the common factor of $45000$ .

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Step 7.3.1

Move the leading negative in $- \frac{1}{216000000}$ into the numerator.

$[\begin{array}{c} - \frac{1}{8000} & \frac{- 1}{216000000} \cdot - 45000 \\ - \frac{1}{216000000} \cdot - 18000 & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.3.2

Factor $45000$ out of $216000000$ .

$[\begin{array}{c} - \frac{1}{8000} & \frac{- 1}{45000 (4800)} \cdot - 45000 \\ - \frac{1}{216000000} \cdot - 18000 & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.3.3

Factor $45000$ out of $- 45000$ .

$[\begin{array}{c} - \frac{1}{8000} & \frac{- 1}{45000 \cdot 4800} \cdot (45000 \cdot - 1) \\ - \frac{1}{216000000} \cdot - 18000 & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.3.4

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{8000} & \frac{- 1}{45000 \cdot 4800} \cdot (45000 \cdot - 1) \\ - \frac{1}{216000000} \cdot - 18000 & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.3.5

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{8000} & \frac{- 1}{4800} \cdot - 1 \\ - \frac{1}{216000000} \cdot - 18000 & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.4

Combine $\frac{- 1}{4800}$ and $- 1$ .

$[\begin{array}{c} - \frac{1}{8000} & \frac{- - 1}{4800} \\ - \frac{1}{216000000} \cdot - 18000 & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.5

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} - \frac{1}{8000} & \frac{1}{4800} \\ - \frac{1}{216000000} \cdot - 18000 & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.6

Cancel the common factor of $18000$ .

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Step 7.6.1

Move the leading negative in $- \frac{1}{216000000}$ into the numerator.

$[\begin{array}{c} - \frac{1}{8000} & \frac{1}{4800} \\ \frac{- 1}{216000000} \cdot - 18000 & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.6.2

Factor $18000$ out of $216000000$ .

$[\begin{array}{c} - \frac{1}{8000} & \frac{1}{4800} \\ \frac{- 1}{18000 (12000)} \cdot - 18000 & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.6.3

Factor $18000$ out of $- 18000$ .

$[\begin{array}{c} - \frac{1}{8000} & \frac{1}{4800} \\ \frac{- 1}{18000 \cdot 12000} \cdot (18000 \cdot - 1) & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.6.4

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{8000} & \frac{1}{4800} \\ \frac{- 1}{18000 \cdot 12000} \cdot (18000 \cdot - 1) & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.6.5

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{8000} & \frac{1}{4800} \\ \frac{- 1}{12000} \cdot - 1 & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.7

Combine $\frac{- 1}{12000}$ and $- 1$ .

$[\begin{array}{c} - \frac{1}{8000} & \frac{1}{4800} \\ \frac{- - 1}{12000} & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.8

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} - \frac{1}{8000} & \frac{1}{4800} \\ \frac{1}{12000} & - \frac{1}{216000000} \cdot 22000 \end{array}]$

Step 7.9

Cancel the common factor of $2000$ .

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Step 7.9.1

Move the leading negative in $- \frac{1}{216000000}$ into the numerator.

$[\begin{array}{c} - \frac{1}{8000} & \frac{1}{4800} \\ \frac{1}{12000} & \frac{- 1}{216000000} \cdot 22000 \end{array}]$

Step 7.9.2

Factor $2000$ out of $216000000$ .

$[\begin{array}{c} - \frac{1}{8000} & \frac{1}{4800} \\ \frac{1}{12000} & \frac{- 1}{2000 (108000)} \cdot 22000 \end{array}]$

Step 7.9.3

Factor $2000$ out of $22000$ .

$[\begin{array}{c} - \frac{1}{8000} & \frac{1}{4800} \\ \frac{1}{12000} & \frac{- 1}{2000 \cdot 108000} \cdot (2000 \cdot 11) \end{array}]$

Step 7.9.4

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{8000} & \frac{1}{4800} \\ \frac{1}{12000} & \frac{- 1}{2000 \cdot 108000} \cdot (2000 \cdot 11) \end{array}]$

Step 7.9.5

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{8000} & \frac{1}{4800} \\ \frac{1}{12000} & \frac{- 1}{108000} \cdot 11 \end{array}]$

Step 7.10

Combine $\frac{- 1}{108000}$ and $11$ .

$[\begin{array}{c} - \frac{1}{8000} & \frac{1}{4800} \\ \frac{1}{12000} & \frac{- 1 \cdot 11}{108000} \end{array}]$

Step 7.11

Multiply $- 1$ by $11$ .

$[\begin{array}{c} - \frac{1}{8000} & \frac{1}{4800} \\ \frac{1}{12000} & \frac{- 11}{108000} \end{array}]$

Step 7.12

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{8000} & \frac{1}{4800} \\ \frac{1}{12000} & - \frac{11}{108000} \end{array}]$