Finite Math Examples

$[\begin{array}{c} \frac{1}{3} & 4 \\ 2 & - \frac{1}{2} \end{array}] \cdot [\begin{array}{c} - \frac{1}{3} & - 4 \\ - 2 & \frac{1}{2} \end{array}]$

Step 1

Multiply $[\begin{array}{c} \frac{1}{3} & 4 \\ 2 & - \frac{1}{2} \end{array}] [\begin{array}{c} - \frac{1}{3} & - 4 \\ - 2 & \frac{1}{2} \end{array}]$ .

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Step 1.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $2 \times 2$ and the second matrix is $2 \times 2$ .

Step 1.2

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} \frac{1}{3} (- \frac{1}{3}) + 4 \cdot - 2 & \frac{1}{3} \cdot - 4 + 4 (\frac{1}{2}) \\ 2 (- \frac{1}{3}) - \frac{1}{2} \cdot - 2 & 2 \cdot - 4 - \frac{1}{2} \cdot \frac{1}{2} \end{array}]$

Step 1.3

Simplify each element of the matrix by multiplying out all the expressions.

$[\begin{array}{c} - \frac{73}{9} & \frac{2}{3} \\ \frac{1}{3} & - \frac{33}{4} \end{array}]$

Step 2

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 3

Find the determinant.

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Step 3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- \frac{73}{9} (- \frac{33}{4}) - \frac{1}{3} \cdot \frac{2}{3}$

Step 3.2

Simplify the determinant.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Cancel the common factor of $3$ .

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Step 3.2.1.1.1

Move the leading negative in $- \frac{73}{9}$ into the numerator.

$\frac{- 73}{9} (- \frac{33}{4}) - \frac{1}{3} \cdot \frac{2}{3}$

Step 3.2.1.1.2

Move the leading negative in $- \frac{33}{4}$ into the numerator.

$\frac{- 73}{9} \cdot \frac{- 33}{4} - \frac{1}{3} \cdot \frac{2}{3}$

Step 3.2.1.1.3

Factor $3$ out of $9$ .

$\frac{- 73}{3 (3)} \cdot \frac{- 33}{4} - \frac{1}{3} \cdot \frac{2}{3}$

Step 3.2.1.1.4

Factor $3$ out of $- 33$ .

$\frac{- 73}{3 \cdot 3} \cdot \frac{3 \cdot - 11}{4} - \frac{1}{3} \cdot \frac{2}{3}$

Step 3.2.1.1.5

Cancel the common factor.

$\frac{- 73}{3 \cdot 3} \cdot \frac{3 \cdot - 11}{4} - \frac{1}{3} \cdot \frac{2}{3}$

Step 3.2.1.1.6

Rewrite the expression.

$\frac{- 73}{3} \cdot \frac{- 11}{4} - \frac{1}{3} \cdot \frac{2}{3}$

Step 3.2.1.2

Multiply $\frac{- 73}{3}$ by $\frac{- 11}{4}$ .

$\frac{- 73 \cdot - 11}{3 \cdot 4} - \frac{1}{3} \cdot \frac{2}{3}$

Step 3.2.1.3

Multiply $- 73$ by $- 11$ .

$\frac{803}{3 \cdot 4} - \frac{1}{3} \cdot \frac{2}{3}$

Step 3.2.1.4

Multiply $3$ by $4$ .

$\frac{803}{12} - \frac{1}{3} \cdot \frac{2}{3}$

Step 3.2.1.5

Multiply $- \frac{1}{3} \cdot \frac{2}{3}$ .

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Step 3.2.1.5.1

Multiply $\frac{2}{3}$ by $\frac{1}{3}$ .

$\frac{803}{12} - \frac{2}{3 \cdot 3}$

Step 3.2.1.5.2

Multiply $3$ by $3$ .

$\frac{803}{12} - \frac{2}{9}$

Step 3.2.2

To write $\frac{803}{12}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{803}{12} \cdot \frac{3}{3} - \frac{2}{9}$

Step 3.2.3

To write $- \frac{2}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{803}{12} \cdot \frac{3}{3} - \frac{2}{9} \cdot \frac{4}{4}$

Step 3.2.4

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.4.1

Multiply $\frac{803}{12}$ by $\frac{3}{3}$ .

$\frac{803 \cdot 3}{12 \cdot 3} - \frac{2}{9} \cdot \frac{4}{4}$

Step 3.2.4.2

Multiply $12$ by $3$ .

$\frac{803 \cdot 3}{36} - \frac{2}{9} \cdot \frac{4}{4}$

Step 3.2.4.3

Multiply $\frac{2}{9}$ by $\frac{4}{4}$ .

$\frac{803 \cdot 3}{36} - \frac{2 \cdot 4}{9 \cdot 4}$

Step 3.2.4.4

Multiply $9$ by $4$ .

$\frac{803 \cdot 3}{36} - \frac{2 \cdot 4}{36}$

Step 3.2.5

Combine the numerators over the common denominator.

$\frac{803 \cdot 3 - 2 \cdot 4}{36}$

Step 3.2.6

Simplify the numerator.

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Step 3.2.6.1

Multiply $803$ by $3$ .

$\frac{2409 - 2 \cdot 4}{36}$

Step 3.2.6.2

Multiply $- 2$ by $4$ .

$\frac{2409 - 8}{36}$

Step 3.2.6.3

Subtract $8$ from $2409$ .

$\frac{2401}{36}$

Step 4

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

Step 5

Substitute the known values into the formula for the inverse.

$\frac{1}{\frac{2401}{36}} [\begin{array}{c} - \frac{33}{4} & - \frac{2}{3} \\ - \frac{1}{3} & - \frac{73}{9} \end{array}]$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{36}{2401}) [\begin{array}{c} - \frac{33}{4} & - \frac{2}{3} \\ - \frac{1}{3} & - \frac{73}{9} \end{array}]$

Step 7

Multiply $\frac{36}{2401}$ by $1$ .

$\frac{36}{2401} [\begin{array}{c} - \frac{33}{4} & - \frac{2}{3} \\ - \frac{1}{3} & - \frac{73}{9} \end{array}]$

Step 8

Multiply $\frac{36}{2401}$ by each element of the matrix.

$[\begin{array}{c} \frac{36}{2401} (- \frac{33}{4}) & \frac{36}{2401} (- \frac{2}{3}) \\ \frac{36}{2401} (- \frac{1}{3}) & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9

Simplify each element in the matrix.

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Step 9.1

Cancel the common factor of $4$ .

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Step 9.1.1

Move the leading negative in $- \frac{33}{4}$ into the numerator.

$[\begin{array}{c} \frac{36}{2401} \cdot \frac{- 33}{4} & \frac{36}{2401} (- \frac{2}{3}) \\ \frac{36}{2401} (- \frac{1}{3}) & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.1.2

Factor $4$ out of $36$ .

$[\begin{array}{c} \frac{4 (9)}{2401} \cdot \frac{- 33}{4} & \frac{36}{2401} (- \frac{2}{3}) \\ \frac{36}{2401} (- \frac{1}{3}) & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.1.3

Cancel the common factor.

$[\begin{array}{c} \frac{4 \cdot 9}{2401} \cdot \frac{- 33}{4} & \frac{36}{2401} (- \frac{2}{3}) \\ \frac{36}{2401} (- \frac{1}{3}) & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.1.4

Rewrite the expression.

$[\begin{array}{c} \frac{9}{2401} \cdot - 33 & \frac{36}{2401} (- \frac{2}{3}) \\ \frac{36}{2401} (- \frac{1}{3}) & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.2

Combine $\frac{9}{2401}$ and $- 33$ .

$[\begin{array}{c} \frac{9 \cdot - 33}{2401} & \frac{36}{2401} (- \frac{2}{3}) \\ \frac{36}{2401} (- \frac{1}{3}) & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.3

Multiply $9$ by $- 33$ .

$[\begin{array}{c} \frac{- 297}{2401} & \frac{36}{2401} (- \frac{2}{3}) \\ \frac{36}{2401} (- \frac{1}{3}) & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.4

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{297}{2401} & \frac{36}{2401} (- \frac{2}{3}) \\ \frac{36}{2401} (- \frac{1}{3}) & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.5

Cancel the common factor of $3$ .

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Step 9.5.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$[\begin{array}{c} - \frac{297}{2401} & \frac{36}{2401} \cdot \frac{- 2}{3} \\ \frac{36}{2401} (- \frac{1}{3}) & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.5.2

Factor $3$ out of $36$ .

$[\begin{array}{c} - \frac{297}{2401} & \frac{3 (12)}{2401} \cdot \frac{- 2}{3} \\ \frac{36}{2401} (- \frac{1}{3}) & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.5.3

Cancel the common factor.

$[\begin{array}{c} - \frac{297}{2401} & \frac{3 \cdot 12}{2401} \cdot \frac{- 2}{3} \\ \frac{36}{2401} (- \frac{1}{3}) & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.5.4

Rewrite the expression.

$[\begin{array}{c} - \frac{297}{2401} & \frac{12}{2401} \cdot - 2 \\ \frac{36}{2401} (- \frac{1}{3}) & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.6

Combine $\frac{12}{2401}$ and $- 2$ .

$[\begin{array}{c} - \frac{297}{2401} & \frac{12 \cdot - 2}{2401} \\ \frac{36}{2401} (- \frac{1}{3}) & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.7

Multiply $12$ by $- 2$ .

$[\begin{array}{c} - \frac{297}{2401} & \frac{- 24}{2401} \\ \frac{36}{2401} (- \frac{1}{3}) & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.8

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{297}{2401} & - \frac{24}{2401} \\ \frac{36}{2401} (- \frac{1}{3}) & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.9

Cancel the common factor of $3$ .

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Step 9.9.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$[\begin{array}{c} - \frac{297}{2401} & - \frac{24}{2401} \\ \frac{36}{2401} \cdot \frac{- 1}{3} & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.9.2

Factor $3$ out of $36$ .

$[\begin{array}{c} - \frac{297}{2401} & - \frac{24}{2401} \\ \frac{3 (12)}{2401} \cdot \frac{- 1}{3} & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.9.3

Cancel the common factor.

$[\begin{array}{c} - \frac{297}{2401} & - \frac{24}{2401} \\ \frac{3 \cdot 12}{2401} \cdot \frac{- 1}{3} & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.9.4

Rewrite the expression.

$[\begin{array}{c} - \frac{297}{2401} & - \frac{24}{2401} \\ \frac{12}{2401} \cdot - 1 & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.10

Combine $\frac{12}{2401}$ and $- 1$ .

$[\begin{array}{c} - \frac{297}{2401} & - \frac{24}{2401} \\ \frac{12 \cdot - 1}{2401} & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.11

Multiply $12$ by $- 1$ .

$[\begin{array}{c} - \frac{297}{2401} & - \frac{24}{2401} \\ \frac{- 12}{2401} & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.12

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{297}{2401} & - \frac{24}{2401} \\ - \frac{12}{2401} & \frac{36}{2401} (- \frac{73}{9}) \end{array}]$

Step 9.13

Cancel the common factor of $9$ .

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Step 9.13.1

Move the leading negative in $- \frac{73}{9}$ into the numerator.

$[\begin{array}{c} - \frac{297}{2401} & - \frac{24}{2401} \\ - \frac{12}{2401} & \frac{36}{2401} \cdot \frac{- 73}{9} \end{array}]$

Step 9.13.2

Factor $9$ out of $36$ .

$[\begin{array}{c} - \frac{297}{2401} & - \frac{24}{2401} \\ - \frac{12}{2401} & \frac{9 (4)}{2401} \cdot \frac{- 73}{9} \end{array}]$

Step 9.13.3

Cancel the common factor.

$[\begin{array}{c} - \frac{297}{2401} & - \frac{24}{2401} \\ - \frac{12}{2401} & \frac{9 \cdot 4}{2401} \cdot \frac{- 73}{9} \end{array}]$

Step 9.13.4

Rewrite the expression.

$[\begin{array}{c} - \frac{297}{2401} & - \frac{24}{2401} \\ - \frac{12}{2401} & \frac{4}{2401} \cdot - 73 \end{array}]$

Step 9.14

Combine $\frac{4}{2401}$ and $- 73$ .

$[\begin{array}{c} - \frac{297}{2401} & - \frac{24}{2401} \\ - \frac{12}{2401} & \frac{4 \cdot - 73}{2401} \end{array}]$

Step 9.15

Multiply $4$ by $- 73$ .

$[\begin{array}{c} - \frac{297}{2401} & - \frac{24}{2401} \\ - \frac{12}{2401} & \frac{- 292}{2401} \end{array}]$

Step 9.16

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{297}{2401} & - \frac{24}{2401} \\ - \frac{12}{2401} & - \frac{292}{2401} \end{array}]$