Finite Math Examples

$- 3 A = [\begin{array}{c} - 9 & - 3 \\ 9 & 8 \end{array}]$

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.

Tap for more steps...

Step 2.1

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

$- 9 \cdot 8 - 9 \cdot - 3$

Step 2.2

Simplify the determinant.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Multiply $- 9$ by $8$ .

$- 72 - 9 \cdot - 3$

Step 2.2.1.2

Multiply $- 9$ by $- 3$ .

$- 72 + 27$

Step 2.2.2

Add $- 72$ and $27$ .

$- 45$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- 45} [\begin{array}{c} 8 & 3 \\ - 9 & - 9 \end{array}]$

Step 5

Move the negative in front of the fraction.

$- \frac{1}{45} [\begin{array}{c} 8 & 3 \\ - 9 & - 9 \end{array}]$

Step 6

Multiply $- \frac{1}{45}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{45} \cdot 8 & - \frac{1}{45} \cdot 3 \\ - \frac{1}{45} \cdot - 9 & - \frac{1}{45} \cdot - 9 \end{array}]$

Step 7

Simplify each element in the matrix.

Tap for more steps...

Step 7.1

Multiply $- \frac{1}{45} \cdot 8$ .

Tap for more steps...

Step 7.1.1

Multiply $8$ by $- 1$ .

$[\begin{array}{c} - 8 (\frac{1}{45}) & - \frac{1}{45} \cdot 3 \\ - \frac{1}{45} \cdot - 9 & - \frac{1}{45} \cdot - 9 \end{array}]$

Step 7.1.2

Combine $- 8$ and $\frac{1}{45}$ .

$[\begin{array}{c} \frac{- 8}{45} & - \frac{1}{45} \cdot 3 \\ - \frac{1}{45} \cdot - 9 & - \frac{1}{45} \cdot - 9 \end{array}]$

Step 7.2

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{8}{45} & - \frac{1}{45} \cdot 3 \\ - \frac{1}{45} \cdot - 9 & - \frac{1}{45} \cdot - 9 \end{array}]$

Step 7.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 7.3.1

Move the leading negative in $- \frac{1}{45}$ into the numerator.

$[\begin{array}{c} - \frac{8}{45} & \frac{- 1}{45} \cdot 3 \\ - \frac{1}{45} \cdot - 9 & - \frac{1}{45} \cdot - 9 \end{array}]$

Step 7.3.2

Factor $3$ out of $45$ .

$[\begin{array}{c} - \frac{8}{45} & \frac{- 1}{3 (15)} \cdot 3 \\ - \frac{1}{45} \cdot - 9 & - \frac{1}{45} \cdot - 9 \end{array}]$

Step 7.3.3

Cancel the common factor.

$[\begin{array}{c} - \frac{8}{45} & \frac{- 1}{3 \cdot 15} \cdot 3 \\ - \frac{1}{45} \cdot - 9 & - \frac{1}{45} \cdot - 9 \end{array}]$

Step 7.3.4

Rewrite the expression.

$[\begin{array}{c} - \frac{8}{45} & \frac{- 1}{15} \\ - \frac{1}{45} \cdot - 9 & - \frac{1}{45} \cdot - 9 \end{array}]$

Step 7.4

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{8}{45} & - \frac{1}{15} \\ - \frac{1}{45} \cdot - 9 & - \frac{1}{45} \cdot - 9 \end{array}]$

Step 7.5

Cancel the common factor of $9$ .

Tap for more steps...

Step 7.5.1

Move the leading negative in $- \frac{1}{45}$ into the numerator.

$[\begin{array}{c} - \frac{8}{45} & - \frac{1}{15} \\ \frac{- 1}{45} \cdot - 9 & - \frac{1}{45} \cdot - 9 \end{array}]$

Step 7.5.2

Factor $9$ out of $45$ .

$[\begin{array}{c} - \frac{8}{45} & - \frac{1}{15} \\ \frac{- 1}{9 (5)} \cdot - 9 & - \frac{1}{45} \cdot - 9 \end{array}]$

Step 7.5.3

Factor $9$ out of $- 9$ .

$[\begin{array}{c} - \frac{8}{45} & - \frac{1}{15} \\ \frac{- 1}{9 \cdot 5} \cdot (9 \cdot - 1) & - \frac{1}{45} \cdot - 9 \end{array}]$

Step 7.5.4

Cancel the common factor.

$[\begin{array}{c} - \frac{8}{45} & - \frac{1}{15} \\ \frac{- 1}{9 \cdot 5} \cdot (9 \cdot - 1) & - \frac{1}{45} \cdot - 9 \end{array}]$

Step 7.5.5

Rewrite the expression.

$[\begin{array}{c} - \frac{8}{45} & - \frac{1}{15} \\ \frac{- 1}{5} \cdot - 1 & - \frac{1}{45} \cdot - 9 \end{array}]$

Step 7.6

Combine $\frac{- 1}{5}$ and $- 1$ .

$[\begin{array}{c} - \frac{8}{45} & - \frac{1}{15} \\ \frac{- - 1}{5} & - \frac{1}{45} \cdot - 9 \end{array}]$

Step 7.7

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} - \frac{8}{45} & - \frac{1}{15} \\ \frac{1}{5} & - \frac{1}{45} \cdot - 9 \end{array}]$

Step 7.8

Cancel the common factor of $9$ .

Tap for more steps...

Step 7.8.1

Move the leading negative in $- \frac{1}{45}$ into the numerator.

$[\begin{array}{c} - \frac{8}{45} & - \frac{1}{15} \\ \frac{1}{5} & \frac{- 1}{45} \cdot - 9 \end{array}]$

Step 7.8.2

Factor $9$ out of $45$ .

$[\begin{array}{c} - \frac{8}{45} & - \frac{1}{15} \\ \frac{1}{5} & \frac{- 1}{9 (5)} \cdot - 9 \end{array}]$

Step 7.8.3

Factor $9$ out of $- 9$ .

$[\begin{array}{c} - \frac{8}{45} & - \frac{1}{15} \\ \frac{1}{5} & \frac{- 1}{9 \cdot 5} \cdot (9 \cdot - 1) \end{array}]$

Step 7.8.4

Cancel the common factor.

$[\begin{array}{c} - \frac{8}{45} & - \frac{1}{15} \\ \frac{1}{5} & \frac{- 1}{9 \cdot 5} \cdot (9 \cdot - 1) \end{array}]$

Step 7.8.5

Rewrite the expression.

$[\begin{array}{c} - \frac{8}{45} & - \frac{1}{15} \\ \frac{1}{5} & \frac{- 1}{5} \cdot - 1 \end{array}]$

Step 7.9

Combine $\frac{- 1}{5}$ and $- 1$ .

$[\begin{array}{c} - \frac{8}{45} & - \frac{1}{15} \\ \frac{1}{5} & \frac{- - 1}{5} \end{array}]$

Step 7.10

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} - \frac{8}{45} & - \frac{1}{15} \\ \frac{1}{5} & \frac{1}{5} \end{array}]$