Finite Math Examples

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

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

$2 \cdot - 1 - (- 4 \cdot 1)$

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 $2$ by $- 1$ .

$- 2 - (- 4 \cdot 1)$

Step 2.2.1.2

Multiply $- (- 4 \cdot 1)$ .

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Step 2.2.1.2.1

Multiply $- 4$ by $1$ .

$- 2 - - 4$

Step 2.2.1.2.2

Multiply $- 1$ by $- 4$ .

$- 2 + 4$

Step 2.2.2

Add $- 2$ and $4$ .

$2$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

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

Step 5

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

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

Step 6

Simplify each element in the matrix.

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Step 6.1

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

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

Step 6.2

Move the negative in front of the fraction.

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

Step 6.3

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

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

Step 6.4

Move the negative in front of the fraction.

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

Step 6.5

Cancel the common factor of $2$ .

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Step 6.5.1

Factor $2$ out of $4$ .

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

Step 6.5.2

Cancel the common factor.

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

Step 6.5.3

Rewrite the expression.

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

Step 6.6

Cancel the common factor of $2$ .

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Step 6.6.1

Cancel the common factor.

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

Step 6.6.2

Rewrite the expression.

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