Precalculus Examples

[\begin{matrix} 4 & 2 3 & 1 \end{matrix}]

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

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{matrix} d & - b - c & a \end{matrix}]

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

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

4 \cdot 1 - 3 \cdot 2

$4 \cdot 1 - 3 \cdot 2$

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

4

$4$ by

1

$1$ .

4 - 3 \cdot 2

$4 - 3 \cdot 2$

Step 2.2.1.2

Multiply

- 3

$- 3$ by

2

$2$ .

4 - 6

$4 - 6$

4 - 6

$4 - 6$

Step 2.2.2

Subtract

6

$6$ from

4

$4$ .

- 2

$- 2$

- 2

$- 2$

- 2

$- 2$

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{matrix} 1 & - 2 - 3 & 4 \end{matrix}]

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

Step 5

Move the negative in front of the fraction.

- \frac{1}{2} [\begin{matrix} 1 & - 2 - 3 & 4 \end{matrix}]

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

Step 6

Multiply

- \frac{1}{2}

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

[\begin{matrix} - \frac{1}{2} \cdot 1 & - \frac{1}{2} \cdot - 2 - \frac{1}{2} \cdot - 3 & - \frac{1}{2} \cdot 4 \end{matrix}]

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

Step 7

Simplify each element in the matrix.

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Step 7.1

Multiply

- 1

$- 1$ by

1

$1$ .

[\begin{matrix} - \frac{1}{2} & - \frac{1}{2} \cdot - 2 - \frac{1}{2} \cdot - 3 & - \frac{1}{2} \cdot 4 \end{matrix}]

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

Step 7.2

Cancel the common factor of

2

$2$ .

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Step 7.2.1

Move the leading negative in

- \frac{1}{2}

$- \frac{1}{2}$ into the numerator.

[\begin{matrix} - \frac{1}{2} & \frac{- 1}{2} \cdot - 2 - \frac{1}{2} \cdot - 3 & - \frac{1}{2} \cdot 4 \end{matrix}]

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

Step 7.2.2

Factor

2

$2$ out of

- 2

$- 2$ .

[\begin{matrix} - \frac{1}{2} & \frac{- 1}{2} \cdot (2 (- 1)) - \frac{1}{2} \cdot - 3 & - \frac{1}{2} \cdot 4 \end{matrix}]

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

Step 7.2.3

Cancel the common factor.

⎡ ⎣ \begin{matrix} - \frac{1}{2} & \frac{- 1}{2} \cdot (2 \cdot - 1) - \frac{1}{2} \cdot - 3 & - \frac{1}{2} \cdot 4 \end{matrix} ⎤ ⎦

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

Step 7.2.4

Rewrite the expression.

[\begin{matrix} - \frac{1}{2} & - 1 \cdot - 1 - \frac{1}{2} \cdot - 3 & - \frac{1}{2} \cdot 4 \end{matrix}]

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

[\begin{matrix} - \frac{1}{2} & - 1 \cdot - 1 - \frac{1}{2} \cdot - 3 & - \frac{1}{2} \cdot 4 \end{matrix}]

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

Step 7.3

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

[\begin{matrix} - \frac{1}{2} & 1 - \frac{1}{2} \cdot - 3 & - \frac{1}{2} \cdot 4 \end{matrix}]

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

Step 7.4

Multiply

- \frac{1}{2} \cdot - 3

$- \frac{1}{2} \cdot - 3$ .

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Step 7.4.1

Multiply

- 3

$- 3$ by

- 1

$- 1$ .

[\begin{matrix} - \frac{1}{2} & 1 3 (\frac{1}{2}) & - \frac{1}{2} \cdot 4 \end{matrix}]

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

Step 7.4.2

Combine

3

$3$ and

\frac{1}{2}

$\frac{1}{2}$ .

[\begin{matrix} - \frac{1}{2} & 1 \frac{3}{2} & - \frac{1}{2} \cdot 4 \end{matrix}]

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

[\begin{matrix} - \frac{1}{2} & 1 \frac{3}{2} & - \frac{1}{2} \cdot 4 \end{matrix}]

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

Step 7.5

Cancel the common factor of

2

$2$ .

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Step 7.5.1

Move the leading negative in

- \frac{1}{2}

$- \frac{1}{2}$ into the numerator.

[\begin{matrix} - \frac{1}{2} & 1 \frac{3}{2} & \frac{- 1}{2} \cdot 4 \end{matrix}]

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

Step 7.5.2

Factor

2

$2$ out of

4

$4$ .

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

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

Step 7.5.3

Cancel the common factor.

⎡ ⎣ \begin{matrix} - \frac{1}{2} & 1 \frac{3}{2} & \frac{- 1}{2} \cdot (2 \cdot 2) \end{matrix} ⎤ ⎦

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

Step 7.5.4

Rewrite the expression.

[\begin{matrix} - \frac{1}{2} & 1 \frac{3}{2} & - 1 \cdot 2 \end{matrix}]

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

[\begin{matrix} - \frac{1}{2} & 1 \frac{3}{2} & - 1 \cdot 2 \end{matrix}]

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

Step 7.6

Multiply

- 1

$- 1$ by

2

$2$ .

[\begin{matrix} - \frac{1}{2} & 1 \frac{3}{2} & - 2 \end{matrix}]

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

[\begin{matrix} - \frac{1}{2} & 1 \frac{3}{2} & - 2 \end{matrix}]

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

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