Precalculus Examples

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

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

$- 1 \cdot 4 - 3 \cdot - 2$

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 $- 1$ by $4$ .

$- 4 - 3 \cdot - 2$

Step 2.2.1.2

Multiply $- 3$ by $- 2$ .

$- 4 + 6$

Step 2.2.2

Add $- 4$ and $6$ .

$2$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

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

Step 5

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

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

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

Factor $2$ out of $4$ .

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

Step 6.1.2

Cancel the common factor.

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

Step 6.1.3

Rewrite the expression.

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

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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

Step 6.3

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

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

Step 6.4

Move the negative in front of the fraction.

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

Step 6.5

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

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

Step 6.6

Move the negative in front of the fraction.

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