Linear Algebra Examples

$[\begin{array}{c} 0 & \frac{1}{8} \\ \frac{1}{14} & \frac{9}{112} \end{array}]$

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

$0 (\frac{9}{112}) - \frac{1}{14} \cdot \frac{1}{8}$

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 $0$ by $\frac{9}{112}$ .

$0 - \frac{1}{14} \cdot \frac{1}{8}$

Step 2.2.1.2

Multiply $- \frac{1}{14} \cdot \frac{1}{8}$ .

Tap for more steps...

Step 2.2.1.2.1

Multiply $\frac{1}{8}$ by $\frac{1}{14}$ .

$0 - \frac{1}{8 \cdot 14}$

Step 2.2.1.2.2

Multiply $8$ by $14$ .

$0 - \frac{1}{112}$

Step 2.2.2

Subtract $\frac{1}{112}$ from $0$ .

$- \frac{1}{112}$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- \frac{1}{112}} [\begin{array}{c} \frac{9}{112} & - \frac{1}{8} \\ - \frac{1}{14} & 0 \end{array}]$

Step 5

Cancel the common factor of $1$ and $- 1$ .

Tap for more steps...

Step 5.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{- \frac{1}{112}} [\begin{array}{c} \frac{9}{112} & - \frac{1}{8} \\ - \frac{1}{14} & 0 \end{array}]$

Step 5.2

Move the negative in front of the fraction.

$- \frac{1}{\frac{1}{112}} [\begin{array}{c} \frac{9}{112} & - \frac{1}{8} \\ - \frac{1}{14} & 0 \end{array}]$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 112) [\begin{array}{c} \frac{9}{112} & - \frac{1}{8} \\ - \frac{1}{14} & 0 \end{array}]$

Step 7

Multiply $- (1 \cdot 112)$ .

Tap for more steps...

Step 7.1

Multiply $112$ by $1$ .

$- 1 \cdot 112 [\begin{array}{c} \frac{9}{112} & - \frac{1}{8} \\ - \frac{1}{14} & 0 \end{array}]$

Step 7.2

Multiply $- 1$ by $112$ .

$- 112 [\begin{array}{c} \frac{9}{112} & - \frac{1}{8} \\ - \frac{1}{14} & 0 \end{array}]$

Step 8

Multiply $- 112$ by each element of the matrix.

$[\begin{array}{c} - 112 (\frac{9}{112}) & - 112 (- \frac{1}{8}) \\ - 112 (- \frac{1}{14}) & - 112 \cdot 0 \end{array}]$

Step 9

Simplify each element in the matrix.

Tap for more steps...

Step 9.1

Cancel the common factor of $112$ .

Tap for more steps...

Step 9.1.1

Factor $112$ out of $- 112$ .

$[\begin{array}{c} 112 (- 1) \frac{9}{112} & - 112 (- \frac{1}{8}) \\ - 112 (- \frac{1}{14}) & - 112 \cdot 0 \end{array}]$

Step 9.1.2

Cancel the common factor.

$[\begin{array}{c} 112 \cdot - 1 \frac{9}{112} & - 112 (- \frac{1}{8}) \\ - 112 (- \frac{1}{14}) & - 112 \cdot 0 \end{array}]$

Step 9.1.3

Rewrite the expression.

$[\begin{array}{c} - 1 \cdot 9 & - 112 (- \frac{1}{8}) \\ - 112 (- \frac{1}{14}) & - 112 \cdot 0 \end{array}]$

Step 9.2

Multiply $- 1$ by $9$ .

$[\begin{array}{c} - 9 & - 112 (- \frac{1}{8}) \\ - 112 (- \frac{1}{14}) & - 112 \cdot 0 \end{array}]$

Step 9.3

Cancel the common factor of $8$ .

Tap for more steps...

Step 9.3.1

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$[\begin{array}{c} - 9 & - 112 (\frac{- 1}{8}) \\ - 112 (- \frac{1}{14}) & - 112 \cdot 0 \end{array}]$

Step 9.3.2

Factor $8$ out of $- 112$ .

$[\begin{array}{c} - 9 & 8 (- 14) \frac{- 1}{8} \\ - 112 (- \frac{1}{14}) & - 112 \cdot 0 \end{array}]$

Step 9.3.3

Cancel the common factor.

$[\begin{array}{c} - 9 & 8 \cdot - 14 \frac{- 1}{8} \\ - 112 (- \frac{1}{14}) & - 112 \cdot 0 \end{array}]$

Step 9.3.4

Rewrite the expression.

$[\begin{array}{c} - 9 & - 14 \cdot - 1 \\ - 112 (- \frac{1}{14}) & - 112 \cdot 0 \end{array}]$

Step 9.4

Multiply $- 14$ by $- 1$ .

$[\begin{array}{c} - 9 & 14 \\ - 112 (- \frac{1}{14}) & - 112 \cdot 0 \end{array}]$

Step 9.5

Cancel the common factor of $14$ .

Tap for more steps...

Step 9.5.1

Move the leading negative in $- \frac{1}{14}$ into the numerator.

$[\begin{array}{c} - 9 & 14 \\ - 112 (\frac{- 1}{14}) & - 112 \cdot 0 \end{array}]$

Step 9.5.2

Factor $14$ out of $- 112$ .

$[\begin{array}{c} - 9 & 14 \\ 14 (- 8) \frac{- 1}{14} & - 112 \cdot 0 \end{array}]$

Step 9.5.3

Cancel the common factor.

$[\begin{array}{c} - 9 & 14 \\ 14 \cdot - 8 \frac{- 1}{14} & - 112 \cdot 0 \end{array}]$

Step 9.5.4

Rewrite the expression.

$[\begin{array}{c} - 9 & 14 \\ - 8 \cdot - 1 & - 112 \cdot 0 \end{array}]$

Step 9.6

Multiply $- 8$ by $- 1$ .

$[\begin{array}{c} - 9 & 14 \\ 8 & - 112 \cdot 0 \end{array}]$

Step 9.7

Multiply $- 112$ by $0$ .

$[\begin{array}{c} - 9 & 14 \\ 8 & 0 \end{array}]$