Finite Math Examples

$[\begin{array}{c} 0.7 & 5 \\ 4 & 1.3 \end{array}]$

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

$0.7 \cdot 1.3 - 4 \cdot 5$

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 $0.7$ by $1.3$ .

$0.91 - 4 \cdot 5$

Step 2.2.1.2

Multiply $- 4$ by $5$ .

$0.91 - 20$

Step 2.2.2

Subtract $20$ from $0.91$ .

$- 19.09$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- 19.09} [\begin{array}{c} 1.3 & - 5 \\ - 4 & 0.7 \end{array}]$

Step 5

Divide $1$ by $- 19.09$ .

$- 0.05238344 [\begin{array}{c} 1.3 & - 5 \\ - 4 & 0.7 \end{array}]$

Step 6

Multiply $- 0.05238344$ by each element of the matrix.

$[\begin{array}{c} - 0.05238344 \cdot 1.3 & - 0.05238344 \cdot - 5 \\ - 0.05238344 \cdot - 4 & - 0.05238344 \cdot 0.7 \end{array}]$

Step 7

Simplify each element in the matrix.

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Step 7.1

Multiply $- 0.05238344$ by $1.3$ .

$[\begin{array}{c} - 0.06809848 & - 0.05238344 \cdot - 5 \\ - 0.05238344 \cdot - 4 & - 0.05238344 \cdot 0.7 \end{array}]$

Step 7.2

Multiply $- 0.05238344$ by $- 5$ .

$[\begin{array}{c} - 0.06809848 & 0.26191723 \\ - 0.05238344 \cdot - 4 & - 0.05238344 \cdot 0.7 \end{array}]$

Step 7.3

Multiply $- 0.05238344$ by $- 4$ .

$[\begin{array}{c} - 0.06809848 & 0.26191723 \\ 0.20953378 & - 0.05238344 \cdot 0.7 \end{array}]$

Step 7.4

Multiply $- 0.05238344$ by $0.7$ .

$[\begin{array}{c} - 0.06809848 & 0.26191723 \\ 0.20953378 & - 0.03666841 \end{array}]$