Finite Math Examples

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

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

$3 \cdot 2 - 4 \cdot 1$

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 $3$ by $2$ .

$6 - 4 \cdot 1$

Step 2.2.1.2

Multiply $- 4$ by $1$ .

$6 - 4$

Step 2.2.2

Subtract $4$ from $6$ .

$2$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

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

Step 5

Multiply $\frac{1}{2}$ by each element of the matrix.

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

Step 6

Simplify each element in the matrix.

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Step 6.1

Cancel the common factor of $2$ .

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

Combine $\frac{1}{2}$ and $- 1$ .

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

Step 6.3

Move the negative in front of the fraction.

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

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Factor $2$ out of $- 4$ .

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

Step 6.4.2

Cancel the common factor.

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

Step 6.4.3

Rewrite the expression.

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

Step 6.5

Combine $\frac{1}{2}$ and $3$ .

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