Find the Inverse [[6,12],[18,7]]

$[\begin{array}{c} 6 & 12 \\ 18 & 7 \end{array}]$

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.

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$6 \cdot 7 - 18 \cdot 12$

Simplify the determinant.

Simplify each term.

Multiply $6$ by $7$ .

$42 - 18 \cdot 12$

Multiply $- 18$ by $12$ .

$42 - 216$

Subtract $216$ from $42$ .

$- 174$

Step 3

Substitute the known values into the formula for the inverse.

$\frac{1}{- 174} [\begin{array}{c} 7 & - 12 \\ - 18 & 6 \end{array}]$

Step 4

Move the negative in front of the fraction.

$- \frac{1}{174} [\begin{array}{c} 7 & - 12 \\ - 18 & 6 \end{array}]$

Step 5

Multiply $- \frac{1}{174}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{174} \cdot 7 & - \frac{1}{174} \cdot - 12 \\ - \frac{1}{174} \cdot - 18 & - \frac{1}{174} \cdot 6 \end{array}]$

Step 6

Simplify each element in the matrix.

Multiply $- \frac{1}{174} \cdot 7$ .

Multiply $7$ by $- 1$ .

$[\begin{array}{c} - 7 (\frac{1}{174}) & - \frac{1}{174} \cdot - 12 \\ - \frac{1}{174} \cdot - 18 & - \frac{1}{174} \cdot 6 \end{array}]$

Combine $- 7$ and $\frac{1}{174}$ .

$[\begin{array}{c} \frac{- 7}{174} & - \frac{1}{174} \cdot - 12 \\ - \frac{1}{174} \cdot - 18 & - \frac{1}{174} \cdot 6 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{7}{174} & - \frac{1}{174} \cdot - 12 \\ - \frac{1}{174} \cdot - 18 & - \frac{1}{174} \cdot 6 \end{array}]$

Cancel the common factor of $6$ .

Move the leading negative in $- \frac{1}{174}$ into the numerator.

$[\begin{array}{c} - \frac{7}{174} & \frac{- 1}{174} \cdot - 12 \\ - \frac{1}{174} \cdot - 18 & - \frac{1}{174} \cdot 6 \end{array}]$

Factor $6$ out of $174$ .

$[\begin{array}{c} - \frac{7}{174} & \frac{- 1}{6 (29)} \cdot - 12 \\ - \frac{1}{174} \cdot - 18 & - \frac{1}{174} \cdot 6 \end{array}]$

Factor $6$ out of $- 12$ .

$[\begin{array}{c} - \frac{7}{174} & \frac{- 1}{6 \cdot 29} \cdot (6 \cdot - 2) \\ - \frac{1}{174} \cdot - 18 & - \frac{1}{174} \cdot 6 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} - \frac{7}{174} & \frac{- 1}{6 \cdot 29} \cdot (6 \cdot - 2) \\ - \frac{1}{174} \cdot - 18 & - \frac{1}{174} \cdot 6 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} - \frac{7}{174} & \frac{- 1}{29} \cdot - 2 \\ - \frac{1}{174} \cdot - 18 & - \frac{1}{174} \cdot 6 \end{array}]$

Combine $\frac{- 1}{29}$ and $- 2$ .

$[\begin{array}{c} - \frac{7}{174} & \frac{- - 2}{29} \\ - \frac{1}{174} \cdot - 18 & - \frac{1}{174} \cdot 6 \end{array}]$

Multiply $- 1$ by $- 2$ .

$[\begin{array}{c} - \frac{7}{174} & \frac{2}{29} \\ - \frac{1}{174} \cdot - 18 & - \frac{1}{174} \cdot 6 \end{array}]$

Cancel the common factor of $6$ .

Move the leading negative in $- \frac{1}{174}$ into the numerator.

$[\begin{array}{c} - \frac{7}{174} & \frac{2}{29} \\ \frac{- 1}{174} \cdot - 18 & - \frac{1}{174} \cdot 6 \end{array}]$

Factor $6$ out of $174$ .

$[\begin{array}{c} - \frac{7}{174} & \frac{2}{29} \\ \frac{- 1}{6 (29)} \cdot - 18 & - \frac{1}{174} \cdot 6 \end{array}]$

Factor $6$ out of $- 18$ .

$[\begin{array}{c} - \frac{7}{174} & \frac{2}{29} \\ \frac{- 1}{6 \cdot 29} \cdot (6 \cdot - 3) & - \frac{1}{174} \cdot 6 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} - \frac{7}{174} & \frac{2}{29} \\ \frac{- 1}{6 \cdot 29} \cdot (6 \cdot - 3) & - \frac{1}{174} \cdot 6 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} - \frac{7}{174} & \frac{2}{29} \\ \frac{- 1}{29} \cdot - 3 & - \frac{1}{174} \cdot 6 \end{array}]$

Combine $\frac{- 1}{29}$ and $- 3$ .

$[\begin{array}{c} - \frac{7}{174} & \frac{2}{29} \\ \frac{- - 3}{29} & - \frac{1}{174} \cdot 6 \end{array}]$

Multiply $- 1$ by $- 3$ .

$[\begin{array}{c} - \frac{7}{174} & \frac{2}{29} \\ \frac{3}{29} & - \frac{1}{174} \cdot 6 \end{array}]$

Cancel the common factor of $6$ .

Move the leading negative in $- \frac{1}{174}$ into the numerator.

$[\begin{array}{c} - \frac{7}{174} & \frac{2}{29} \\ \frac{3}{29} & \frac{- 1}{174} \cdot 6 \end{array}]$

Factor $6$ out of $174$ .

$[\begin{array}{c} - \frac{7}{174} & \frac{2}{29} \\ \frac{3}{29} & \frac{- 1}{6 (29)} \cdot 6 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} - \frac{7}{174} & \frac{2}{29} \\ \frac{3}{29} & \frac{- 1}{6 \cdot 29} \cdot 6 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} - \frac{7}{174} & \frac{2}{29} \\ \frac{3}{29} & \frac{- 1}{29} \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{7}{174} & \frac{2}{29} \\ \frac{3}{29} & - \frac{1}{29} \end{array}]$

Find the Inverse [[6,12],[18,7]]

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