Finite Math Examples

$[\begin{array}{c} \frac{1}{5} & \frac{1}{2} \\ \frac{1}{3} & 0 \end{array}]$

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

$\frac{1}{5} \cdot 0 - \frac{1}{3} \cdot \frac{1}{2}$

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 $\frac{1}{5}$ by $0$ .

$0 - \frac{1}{3} \cdot \frac{1}{2}$

Step 2.2.1.2

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

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Step 2.2.1.2.1

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

$0 - \frac{1}{2 \cdot 3}$

Step 2.2.1.2.2

Multiply $2$ by $3$ .

$0 - \frac{1}{6}$

Step 2.2.2

Subtract $\frac{1}{6}$ from $0$ .

$- \frac{1}{6}$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- \frac{1}{6}} [\begin{array}{c} 0 & - \frac{1}{2} \\ - \frac{1}{3} & \frac{1}{5} \end{array}]$

Step 5

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

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Step 5.1

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

$\frac{- 1 (- 1)}{- \frac{1}{6}} [\begin{array}{c} 0 & - \frac{1}{2} \\ - \frac{1}{3} & \frac{1}{5} \end{array}]$

Step 5.2

Move the negative in front of the fraction.

$- \frac{1}{\frac{1}{6}} [\begin{array}{c} 0 & - \frac{1}{2} \\ - \frac{1}{3} & \frac{1}{5} \end{array}]$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 6) [\begin{array}{c} 0 & - \frac{1}{2} \\ - \frac{1}{3} & \frac{1}{5} \end{array}]$

Step 7

Multiply $- (1 \cdot 6)$ .

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Step 7.1

Multiply $6$ by $1$ .

$- 1 \cdot 6 [\begin{array}{c} 0 & - \frac{1}{2} \\ - \frac{1}{3} & \frac{1}{5} \end{array}]$

Step 7.2

Multiply $- 1$ by $6$ .

$- 6 [\begin{array}{c} 0 & - \frac{1}{2} \\ - \frac{1}{3} & \frac{1}{5} \end{array}]$

Step 8

Multiply $- 6$ by each element of the matrix.

$[\begin{array}{c} - 6 \cdot 0 & - 6 (- \frac{1}{2}) \\ - 6 (- \frac{1}{3}) & - 6 (\frac{1}{5}) \end{array}]$

Step 9

Simplify each element in the matrix.

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Step 9.1

Multiply $- 6$ by $0$ .

$[\begin{array}{c} 0 & - 6 (- \frac{1}{2}) \\ - 6 (- \frac{1}{3}) & - 6 (\frac{1}{5}) \end{array}]$

Step 9.2

Cancel the common factor of $2$ .

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Step 9.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$[\begin{array}{c} 0 & - 6 (\frac{- 1}{2}) \\ - 6 (- \frac{1}{3}) & - 6 (\frac{1}{5}) \end{array}]$

Step 9.2.2

Factor $2$ out of $- 6$ .

$[\begin{array}{c} 0 & 2 (- 3) \frac{- 1}{2} \\ - 6 (- \frac{1}{3}) & - 6 (\frac{1}{5}) \end{array}]$

Step 9.2.3

Cancel the common factor.

$[\begin{array}{c} 0 & 2 \cdot - 3 \frac{- 1}{2} \\ - 6 (- \frac{1}{3}) & - 6 (\frac{1}{5}) \end{array}]$

Step 9.2.4

Rewrite the expression.

$[\begin{array}{c} 0 & - 3 \cdot - 1 \\ - 6 (- \frac{1}{3}) & - 6 (\frac{1}{5}) \end{array}]$

Step 9.3

Multiply $- 3$ by $- 1$ .

$[\begin{array}{c} 0 & 3 \\ - 6 (- \frac{1}{3}) & - 6 (\frac{1}{5}) \end{array}]$

Step 9.4

Cancel the common factor of $3$ .

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Step 9.4.1

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

$[\begin{array}{c} 0 & 3 \\ - 6 (\frac{- 1}{3}) & - 6 (\frac{1}{5}) \end{array}]$

Step 9.4.2

Factor $3$ out of $- 6$ .

$[\begin{array}{c} 0 & 3 \\ 3 (- 2) \frac{- 1}{3} & - 6 (\frac{1}{5}) \end{array}]$

Step 9.4.3

Cancel the common factor.

$[\begin{array}{c} 0 & 3 \\ 3 \cdot - 2 \frac{- 1}{3} & - 6 (\frac{1}{5}) \end{array}]$

Step 9.4.4

Rewrite the expression.

$[\begin{array}{c} 0 & 3 \\ - 2 \cdot - 1 & - 6 (\frac{1}{5}) \end{array}]$

Step 9.5

Multiply $- 2$ by $- 1$ .

$[\begin{array}{c} 0 & 3 \\ 2 & - 6 (\frac{1}{5}) \end{array}]$

Step 9.6

Combine $- 6$ and $\frac{1}{5}$ .

$[\begin{array}{c} 0 & 3 \\ 2 & \frac{- 6}{5} \end{array}]$

Step 9.7

Move the negative in front of the fraction.

$[\begin{array}{c} 0 & 3 \\ 2 & - \frac{6}{5} \end{array}]$