Precalculus Examples

$[\begin{array}{c} 2 & 4 \\ 6 & 8 \end{array}] + [\begin{array}{c} 1 & 2 \\ 3 & 4 \end{array}]$

Step 1

Add the corresponding elements.

$[\begin{array}{c} 2 + 1 & 4 + 2 \\ 6 + 3 & 8 + 4 \end{array}]$

Step 2

Simplify each element.

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Step 2.1

Add $2$ and $1$ .

$[\begin{array}{c} 3 & 4 + 2 \\ 6 + 3 & 8 + 4 \end{array}]$

Step 2.2

Add $4$ and $2$ .

$[\begin{array}{c} 3 & 6 \\ 6 + 3 & 8 + 4 \end{array}]$

Step 2.3

Add $6$ and $3$ .

$[\begin{array}{c} 3 & 6 \\ 9 & 8 + 4 \end{array}]$

Step 2.4

Add $8$ and $4$ .

$[\begin{array}{c} 3 & 6 \\ 9 & 12 \end{array}]$

Step 3

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 4

Find the determinant.

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Step 4.1

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

$3 \cdot 12 - 9 \cdot 6$

Step 4.2

Simplify the determinant.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $3$ by $12$ .

$36 - 9 \cdot 6$

Step 4.2.1.2

Multiply $- 9$ by $6$ .

$36 - 54$

Step 4.2.2

Subtract $54$ from $36$ .

$- 18$

Step 5

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

Step 6

Substitute the known values into the formula for the inverse.

$\frac{1}{- 18} [\begin{array}{c} 12 & - 6 \\ - 9 & 3 \end{array}]$

Step 7

Move the negative in front of the fraction.

$- \frac{1}{18} [\begin{array}{c} 12 & - 6 \\ - 9 & 3 \end{array}]$

Step 8

Multiply $- \frac{1}{18}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{18} \cdot 12 & - \frac{1}{18} \cdot - 6 \\ - \frac{1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9

Simplify each element in the matrix.

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Step 9.1

Cancel the common factor of $6$ .

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Step 9.1.1

Move the leading negative in $- \frac{1}{18}$ into the numerator.

$[\begin{array}{c} \frac{- 1}{18} \cdot 12 & - \frac{1}{18} \cdot - 6 \\ - \frac{1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.1.2

Factor $6$ out of $18$ .

$[\begin{array}{c} \frac{- 1}{6 (3)} \cdot 12 & - \frac{1}{18} \cdot - 6 \\ - \frac{1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.1.3

Factor $6$ out of $12$ .

$[\begin{array}{c} \frac{- 1}{6 \cdot 3} \cdot (6 \cdot 2) & - \frac{1}{18} \cdot - 6 \\ - \frac{1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.1.4

Cancel the common factor.

$[\begin{array}{c} \frac{- 1}{6 \cdot 3} \cdot (6 \cdot 2) & - \frac{1}{18} \cdot - 6 \\ - \frac{1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.1.5

Rewrite the expression.

$[\begin{array}{c} \frac{- 1}{3} \cdot 2 & - \frac{1}{18} \cdot - 6 \\ - \frac{1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.2

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

$[\begin{array}{c} \frac{- 1 \cdot 2}{3} & - \frac{1}{18} \cdot - 6 \\ - \frac{1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.3

Multiply $- 1$ by $2$ .

$[\begin{array}{c} \frac{- 2}{3} & - \frac{1}{18} \cdot - 6 \\ - \frac{1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.4

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{2}{3} & - \frac{1}{18} \cdot - 6 \\ - \frac{1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.5

Cancel the common factor of $6$ .

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Step 9.5.1

Move the leading negative in $- \frac{1}{18}$ into the numerator.

$[\begin{array}{c} - \frac{2}{3} & \frac{- 1}{18} \cdot - 6 \\ - \frac{1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.5.2

Factor $6$ out of $18$ .

$[\begin{array}{c} - \frac{2}{3} & \frac{- 1}{6 (3)} \cdot - 6 \\ - \frac{1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.5.3

Factor $6$ out of $- 6$ .

$[\begin{array}{c} - \frac{2}{3} & \frac{- 1}{6 \cdot 3} \cdot (6 \cdot - 1) \\ - \frac{1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.5.4

Cancel the common factor.

$[\begin{array}{c} - \frac{2}{3} & \frac{- 1}{6 \cdot 3} \cdot (6 \cdot - 1) \\ - \frac{1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.5.5

Rewrite the expression.

$[\begin{array}{c} - \frac{2}{3} & \frac{- 1}{3} \cdot - 1 \\ - \frac{1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.6

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

$[\begin{array}{c} - \frac{2}{3} & \frac{- - 1}{3} \\ - \frac{1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.7

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} - \frac{2}{3} & \frac{1}{3} \\ - \frac{1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.8

Cancel the common factor of $9$ .

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Step 9.8.1

Move the leading negative in $- \frac{1}{18}$ into the numerator.

$[\begin{array}{c} - \frac{2}{3} & \frac{1}{3} \\ \frac{- 1}{18} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.8.2

Factor $9$ out of $18$ .

$[\begin{array}{c} - \frac{2}{3} & \frac{1}{3} \\ \frac{- 1}{9 (2)} \cdot - 9 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.8.3

Factor $9$ out of $- 9$ .

$[\begin{array}{c} - \frac{2}{3} & \frac{1}{3} \\ \frac{- 1}{9 \cdot 2} \cdot (9 \cdot - 1) & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.8.4

Cancel the common factor.

$[\begin{array}{c} - \frac{2}{3} & \frac{1}{3} \\ \frac{- 1}{9 \cdot 2} \cdot (9 \cdot - 1) & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.8.5

Rewrite the expression.

$[\begin{array}{c} - \frac{2}{3} & \frac{1}{3} \\ \frac{- 1}{2} \cdot - 1 & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.9

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

$[\begin{array}{c} - \frac{2}{3} & \frac{1}{3} \\ \frac{- - 1}{2} & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.10

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} - \frac{2}{3} & \frac{1}{3} \\ \frac{1}{2} & - \frac{1}{18} \cdot 3 \end{array}]$

Step 9.11

Cancel the common factor of $3$ .

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Step 9.11.1

Move the leading negative in $- \frac{1}{18}$ into the numerator.

$[\begin{array}{c} - \frac{2}{3} & \frac{1}{3} \\ \frac{1}{2} & \frac{- 1}{18} \cdot 3 \end{array}]$

Step 9.11.2

Factor $3$ out of $18$ .

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

Step 9.11.3

Cancel the common factor.

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

Step 9.11.4

Rewrite the expression.

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

Step 9.12

Move the negative in front of the fraction.

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

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