Finite Math Examples

[\begin{array}{c} 0.8 & - 0.6 \\ - 0.5 & 0.4 \end{array}]

$[\begin{array}{c} 0.8 & - 0.6 \\ - 0.5 & 0.4 \end{array}]$

Step 1

The inverse of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]

$\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where

a d - b c

$a d - b c$ is the determinant.

Step 2

Find the determinant.

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Step 2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

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

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

0.8 \cdot 0.4 - (- 0.5 \cdot - 0.6)

$0.8 \cdot 0.4 - (- 0.5 \cdot - 0.6)$

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

0.8

$0.8$ by

0.4

$0.4$ .

0.32 - (- 0.5 \cdot - 0.6)

$0.32 - (- 0.5 \cdot - 0.6)$

Step 2.2.1.2

Multiply

- (- 0.5 \cdot - 0.6)

$- (- 0.5 \cdot - 0.6)$ .

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Step 2.2.1.2.1

Multiply

- 0.5

$- 0.5$ by

- 0.6

$- 0.6$ .

0.32 - 1 \cdot 0.3

$0.32 - 1 \cdot 0.3$

Step 2.2.1.2.2

Multiply

- 1

$- 1$ by

0.3

$0.3$ .

0.32 - 0.3

$0.32 - 0.3$

0.32 - 0.3

$0.32 - 0.3$

0.32 - 0.3

$0.32 - 0.3$

Step 2.2.2

Subtract

0.3

$0.3$ from

0.32

$0.32$ .

0.02

$0.02$

0.02

$0.02$

0.02

$0.02$

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}{0.02} [\begin{array}{c} 0.4 & 0.6 \\ 0.5 & 0.8 \end{array}]

$\frac{1}{0.02} [\begin{array}{c} 0.4 & 0.6 \\ 0.5 & 0.8 \end{array}]$

Step 5

Divide

1

$1$ by

0.02

$0.02$ .

50 [\begin{array}{c} 0.4 & 0.6 \\ 0.5 & 0.8 \end{array}]

$50 [\begin{array}{c} 0.4 & 0.6 \\ 0.5 & 0.8 \end{array}]$

Step 6

Multiply

50

$50$ by each element of the matrix.

[\begin{array}{c} 50 \cdot 0.4 & 50 \cdot 0.6 \\ 50 \cdot 0.5 & 50 \cdot 0.8 \end{array}]

$[\begin{array}{c} 50 \cdot 0.4 & 50 \cdot 0.6 \\ 50 \cdot 0.5 & 50 \cdot 0.8 \end{array}]$

Step 7

Simplify each element in the matrix.

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Step 7.1

Multiply

50

$50$ by

0.4

$0.4$ .

[\begin{array}{c} 20 & 50 \cdot 0.6 \\ 50 \cdot 0.5 & 50 \cdot 0.8 \end{array}]

$[\begin{array}{c} 20 & 50 \cdot 0.6 \\ 50 \cdot 0.5 & 50 \cdot 0.8 \end{array}]$

Step 7.2

Multiply

50

$50$ by

0.6

$0.6$ .

[\begin{array}{c} 20 & 30 \\ 50 \cdot 0.5 & 50 \cdot 0.8 \end{array}]

$[\begin{array}{c} 20 & 30 \\ 50 \cdot 0.5 & 50 \cdot 0.8 \end{array}]$

Step 7.3

Multiply

50

$50$ by

0.5

$0.5$ .

[\begin{array}{c} 20 & 30 \\ 25 & 50 \cdot 0.8 \end{array}]

$[\begin{array}{c} 20 & 30 \\ 25 & 50 \cdot 0.8 \end{array}]$

Step 7.4

Multiply

50

$50$ by

0.8

$0.8$ .

[\begin{array}{c} 20 & 30 \\ 25 & 40 \end{array}]

$[\begin{array}{c} 20 & 30 \\ 25 & 40 \end{array}]$

[\begin{array}{c} 20 & 30 \\ 25 & 40 \end{array}]

$[\begin{array}{c} 20 & 30 \\ 25 & 40 \end{array}]$