Precalculus Examples

[\begin{matrix} 1 & 3 12 & 2 \end{matrix}]

$[\begin{array}{c} 1 & 3 \\ 12 & 2 \end{array}]$

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

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

1 \cdot 2 - 12 \cdot 3

$1 \cdot 2 - 12 \cdot 3$

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

$2$ by

1

$1$ .

2 - 12 \cdot 3

$2 - 12 \cdot 3$

Step 2.2.1.2

Multiply

- 12

$- 12$ by

3

$3$ .

2 - 36

$2 - 36$

2 - 36

$2 - 36$

Step 2.2.2

Subtract

36

$36$ from

2

$2$ .

- 34

$- 34$

- 34

$- 34$

- 34

$- 34$

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}{- 34} [\begin{matrix} 2 & - 3 - 12 & 1 \end{matrix}]

$\frac{1}{- 34} [\begin{array}{c} 2 & - 3 \\ - 12 & 1 \end{array}]$

Step 5

Move the negative in front of the fraction.

- \frac{1}{34} [\begin{matrix} 2 & - 3 - 12 & 1 \end{matrix}]

$- \frac{1}{34} [\begin{array}{c} 2 & - 3 \\ - 12 & 1 \end{array}]$

Step 6

Multiply

- \frac{1}{34}

$- \frac{1}{34}$ by each element of the matrix.

[\begin{matrix} - \frac{1}{34} \cdot 2 & - \frac{1}{34} \cdot - 3 - \frac{1}{34} \cdot - 12 & - \frac{1}{34} \cdot 1 \end{matrix}]

$[\begin{array}{c} - \frac{1}{34} \cdot 2 & - \frac{1}{34} \cdot - 3 \\ - \frac{1}{34} \cdot - 12 & - \frac{1}{34} \cdot 1 \end{array}]$

Step 7

Simplify each element in the matrix.

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Step 7.1

Cancel the common factor of

2

$2$ .

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Step 7.1.1

Move the leading negative in

- \frac{1}{34}

$- \frac{1}{34}$ into the numerator.

[\begin{matrix} \frac{- 1}{34} \cdot 2 & - \frac{1}{34} \cdot - 3 - \frac{1}{34} \cdot - 12 & - \frac{1}{34} \cdot 1 \end{matrix}]

$[\begin{array}{c} \frac{- 1}{34} \cdot 2 & - \frac{1}{34} \cdot - 3 \\ - \frac{1}{34} \cdot - 12 & - \frac{1}{34} \cdot 1 \end{array}]$

Step 7.1.2

Factor

2

$2$ out of

34

$34$ .

⎡ ⎣ \begin{matrix} \frac{- 1}{2 (17)} \cdot 2 & - \frac{1}{34} \cdot - 3 - \frac{1}{34} \cdot - 12 & - \frac{1}{34} \cdot 1 \end{matrix} ⎤ ⎦

$[\begin{array}{c} \frac{- 1}{2 (17)} \cdot 2 & - \frac{1}{34} \cdot - 3 \\ - \frac{1}{34} \cdot - 12 & - \frac{1}{34} \cdot 1 \end{array}]$

Step 7.1.3

Cancel the common factor.

$[\begin{array}{c} \frac{- 1}{2 \cdot 17} \cdot 2 & - \frac{1}{34} \cdot - 3 \\ - \frac{1}{34} \cdot - 12 & - \frac{1}{34} \cdot 1 \end{array}]$

Step 7.1.4

Rewrite the expression.

$[\begin{array}{c} \frac{- 1}{17} & - \frac{1}{34} \cdot - 3 \\ - \frac{1}{34} \cdot - 12 & - \frac{1}{34} \cdot 1 \end{array}]$

Step 7.2

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{17} & - \frac{1}{34} \cdot - 3 \\ - \frac{1}{34} \cdot - 12 & - \frac{1}{34} \cdot 1 \end{array}]$

Step 7.3

Multiply

$- \frac{1}{34} \cdot - 3$ .

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Step 7.3.1

Multiply

$- 3$ by

$- 1$ .

$[\begin{array}{c} - \frac{1}{17} & 3 (\frac{1}{34}) \\ - \frac{1}{34} \cdot - 12 & - \frac{1}{34} \cdot 1 \end{array}]$

Step 7.3.2

Combine

$3$ and

$\frac{1}{34}$ .

$[\begin{array}{c} - \frac{1}{17} & \frac{3}{34} \\ - \frac{1}{34} \cdot - 12 & - \frac{1}{34} \cdot 1 \end{array}]$

Step 7.4

Cancel the common factor of

$2$ .

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Step 7.4.1

Move the leading negative in

$- \frac{1}{34}$ into the numerator.

$[\begin{array}{c} - \frac{1}{17} & \frac{3}{34} \\ \frac{- 1}{34} \cdot - 12 & - \frac{1}{34} \cdot 1 \end{array}]$

Step 7.4.2

Factor

$2$ out of

$34$ .

$[\begin{array}{c} - \frac{1}{17} & \frac{3}{34} \\ \frac{- 1}{2 (17)} \cdot - 12 & - \frac{1}{34} \cdot 1 \end{array}]$

Step 7.4.3

Factor

$2$ out of

$- 12$ .

$[\begin{array}{c} - \frac{1}{17} & \frac{3}{34} \\ \frac{- 1}{2 \cdot 17} \cdot (2 \cdot - 6) & - \frac{1}{34} \cdot 1 \end{array}]$

Step 7.4.4

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{17} & \frac{3}{34} \\ \frac{- 1}{2 \cdot 17} \cdot (2 \cdot - 6) & - \frac{1}{34} \cdot 1 \end{array}]$

Step 7.4.5

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{17} & \frac{3}{34} \\ \frac{- 1}{17} \cdot - 6 & - \frac{1}{34} \cdot 1 \end{array}]$

Step 7.5

Combine

$\frac{- 1}{17}$ and

$- 6$ .

$[\begin{array}{c} - \frac{1}{17} & \frac{3}{34} \\ \frac{- - 6}{17} & - \frac{1}{34} \cdot 1 \end{array}]$

Step 7.6

Multiply

$- 1$ by

$- 6$ .

$[\begin{array}{c} - \frac{1}{17} & \frac{3}{34} \\ \frac{6}{17} & - \frac{1}{34} \cdot 1 \end{array}]$

Step 7.7

Multiply

$- 1$ by

$1$ .

$[\begin{array}{c} - \frac{1}{17} & \frac{3}{34} \\ \frac{6}{17} & - \frac{1}{34} \end{array}]$

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