Linear Algebra Examples

[\begin{matrix} 10 & 9 - 6 & - 5 \end{matrix}]

$[\begin{array}{c} 10 & 9 \\ - 6 & - 5 \end{array}]$

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.

Tap for more steps...

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

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

10 \cdot - 5 - (- 6 \cdot 9)

$10 \cdot - 5 - (- 6 \cdot 9)$

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

10

$10$ by

- 5

$- 5$ .

- 50 - (- 6 \cdot 9)

$- 50 - (- 6 \cdot 9)$

Step 2.2.1.2

Multiply

- (- 6 \cdot 9)

$- (- 6 \cdot 9)$ .

Tap for more steps...

Step 2.2.1.2.1

Multiply

- 6

$- 6$ by

9

$9$ .

- 50 - - 54

$- 50 - - 54$

Step 2.2.1.2.2

Multiply

- 1

$- 1$ by

- 54

$- 54$ .

- 50 + 54

$- 50 + 54$

- 50 + 54

$- 50 + 54$

- 50 + 54

$- 50 + 54$

Step 2.2.2

Add

- 50

$- 50$ and

54

$54$ .

4

$4$

4

$4$

4

$4$

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}{4} [\begin{matrix} - 5 & - 9 6 & 10 \end{matrix}]

$\frac{1}{4} [\begin{array}{c} - 5 & - 9 \\ 6 & 10 \end{array}]$

Step 5

Multiply

\frac{1}{4}

$\frac{1}{4}$ by each element of the matrix.

[\begin{matrix} \frac{1}{4} \cdot - 5 & \frac{1}{4} \cdot - 9 \frac{1}{4} \cdot 6 & \frac{1}{4} \cdot 10 \end{matrix}]

$[\begin{array}{c} \frac{1}{4} \cdot - 5 & \frac{1}{4} \cdot - 9 \\ \frac{1}{4} \cdot 6 & \frac{1}{4} \cdot 10 \end{array}]$

Step 6

Simplify each element in the matrix.

Tap for more steps...

Step 6.1

Combine

\frac{1}{4}

$\frac{1}{4}$ and

- 5

$- 5$ .

[\begin{matrix} \frac{- 5}{4} & \frac{1}{4} \cdot - 9 \frac{1}{4} \cdot 6 & \frac{1}{4} \cdot 10 \end{matrix}]

$[\begin{array}{c} \frac{- 5}{4} & \frac{1}{4} \cdot - 9 \\ \frac{1}{4} \cdot 6 & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.2

Move the negative in front of the fraction.

[\begin{matrix} - \frac{5}{4} & \frac{1}{4} \cdot - 9 \frac{1}{4} \cdot 6 & \frac{1}{4} \cdot 10 \end{matrix}]

$[\begin{array}{c} - \frac{5}{4} & \frac{1}{4} \cdot - 9 \\ \frac{1}{4} \cdot 6 & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.3

Combine

\frac{1}{4}

$\frac{1}{4}$ and

- 9

$- 9$ .

[\begin{matrix} - \frac{5}{4} & \frac{- 9}{4} \frac{1}{4} \cdot 6 & \frac{1}{4} \cdot 10 \end{matrix}]

$[\begin{array}{c} - \frac{5}{4} & \frac{- 9}{4} \\ \frac{1}{4} \cdot 6 & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.4

Move the negative in front of the fraction.

[\begin{matrix} - \frac{5}{4} & - \frac{9}{4} \frac{1}{4} \cdot 6 & \frac{1}{4} \cdot 10 \end{matrix}]

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{1}{4} \cdot 6 & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.5

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 6.5.1

Factor

2

$2$ out of

4

$4$ .

⎡ ⎣ \begin{matrix} - \frac{5}{4} & - \frac{9}{4} \frac{1}{2 (2)} \cdot 6 & \frac{1}{4} \cdot 10 \end{matrix} ⎤ ⎦

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{1}{2 (2)} \cdot 6 & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.5.2

Factor

2

$2$ out of

6

$6$ .

[\begin{matrix} - \frac{5}{4} & - \frac{9}{4} \frac{1}{2 \cdot 2} \cdot (2 \cdot 3) & \frac{1}{4} \cdot 10 \end{matrix}]

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{1}{2 \cdot 2} \cdot (2 \cdot 3) & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.5.3

Cancel the common factor.

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{1}{2 \cdot 2} \cdot (2 \cdot 3) & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.5.4

Rewrite the expression.

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{1}{2} \cdot 3 & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.6

Combine

$\frac{1}{2}$ and

$3$ .

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{3}{2} & \frac{1}{4} \cdot 10 \end{array}]$

Step 6.7

Cancel the common factor of

$2$ .

Tap for more steps...

Step 6.7.1

Factor

$2$ out of

$4$ .

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{3}{2} & \frac{1}{2 (2)} \cdot 10 \end{array}]$

Step 6.7.2

Factor

$2$ out of

$10$ .

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{3}{2} & \frac{1}{2 \cdot 2} \cdot (2 \cdot 5) \end{array}]$

Step 6.7.3

Cancel the common factor.

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{3}{2} & \frac{1}{2 \cdot 2} \cdot (2 \cdot 5) \end{array}]$

Step 6.7.4

Rewrite the expression.

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{3}{2} & \frac{1}{2} \cdot 5 \end{array}]$

Step 6.8

Combine

$\frac{1}{2}$ and

$5$ .

$[\begin{array}{c} - \frac{5}{4} & - \frac{9}{4} \\ \frac{3}{2} & \frac{5}{2} \end{array}]$