Linear Algebra Examples

$[\begin{array}{c} - 1 & \frac{7}{3} & - 2 \\ 0 & \frac{2}{3} & 1 \\ \frac{1}{2} & - 1 & 1 \end{array}] - [\begin{array}{c} 1 & - 10 & 2 \\ - 9 & - 2 & - 4 \\ 5 & 4 & 4 \end{array}]$

Step 1

Subtract the corresponding elements.

$[\begin{array}{c} - 1 - 1 & \frac{7}{3} + 10 & - 2 - 2 \\ 0 + 9 & \frac{2}{3} + 2 & 1 + 4 \\ \frac{1}{2} - 5 & - 1 - 4 & 1 - 4 \end{array}]$

Step 2

Simplify each element.

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Step 2.1

Subtract $1$ from $- 1$ .

$[\begin{array}{c} - 2 & \frac{7}{3} + 10 & - 2 - 2 \\ 0 + 9 & \frac{2}{3} + 2 & 1 + 4 \\ \frac{1}{2} - 5 & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.2

To write $10$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$[\begin{array}{c} - 2 & \frac{7}{3} + 10 \cdot \frac{3}{3} & - 2 - 2 \\ 0 + 9 & \frac{2}{3} + 2 & 1 + 4 \\ \frac{1}{2} - 5 & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.3

Combine $10$ and $\frac{3}{3}$ .

$[\begin{array}{c} - 2 & \frac{7}{3} + \frac{10 \cdot 3}{3} & - 2 - 2 \\ 0 + 9 & \frac{2}{3} + 2 & 1 + 4 \\ \frac{1}{2} - 5 & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.4

Combine the numerators over the common denominator.

$[\begin{array}{c} - 2 & \frac{7 + 10 \cdot 3}{3} & - 2 - 2 \\ 0 + 9 & \frac{2}{3} + 2 & 1 + 4 \\ \frac{1}{2} - 5 & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $10$ by $3$ .

$[\begin{array}{c} - 2 & \frac{7 + 30}{3} & - 2 - 2 \\ 0 + 9 & \frac{2}{3} + 2 & 1 + 4 \\ \frac{1}{2} - 5 & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.5.2

Add $7$ and $30$ .

$[\begin{array}{c} - 2 & \frac{37}{3} & - 2 - 2 \\ 0 + 9 & \frac{2}{3} + 2 & 1 + 4 \\ \frac{1}{2} - 5 & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.6

Subtract $2$ from $- 2$ .

$[\begin{array}{c} - 2 & \frac{37}{3} & - 4 \\ 0 + 9 & \frac{2}{3} + 2 & 1 + 4 \\ \frac{1}{2} - 5 & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.7

Add $0$ and $9$ .

$[\begin{array}{c} - 2 & \frac{37}{3} & - 4 \\ 9 & \frac{2}{3} + 2 & 1 + 4 \\ \frac{1}{2} - 5 & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.8

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$[\begin{array}{c} - 2 & \frac{37}{3} & - 4 \\ 9 & \frac{2}{3} + 2 \cdot \frac{3}{3} & 1 + 4 \\ \frac{1}{2} - 5 & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.9

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

$[\begin{array}{c} - 2 & \frac{37}{3} & - 4 \\ 9 & \frac{2}{3} + \frac{2 \cdot 3}{3} & 1 + 4 \\ \frac{1}{2} - 5 & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.10

Combine the numerators over the common denominator.

$[\begin{array}{c} - 2 & \frac{37}{3} & - 4 \\ 9 & \frac{2 + 2 \cdot 3}{3} & 1 + 4 \\ \frac{1}{2} - 5 & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.11

Simplify the numerator.

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Step 2.11.1

Multiply $2$ by $3$ .

$[\begin{array}{c} - 2 & \frac{37}{3} & - 4 \\ 9 & \frac{2 + 6}{3} & 1 + 4 \\ \frac{1}{2} - 5 & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.11.2

Add $2$ and $6$ .

$[\begin{array}{c} - 2 & \frac{37}{3} & - 4 \\ 9 & \frac{8}{3} & 1 + 4 \\ \frac{1}{2} - 5 & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.12

Add $1$ and $4$ .

$[\begin{array}{c} - 2 & \frac{37}{3} & - 4 \\ 9 & \frac{8}{3} & 5 \\ \frac{1}{2} - 5 & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.13

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$[\begin{array}{c} - 2 & \frac{37}{3} & - 4 \\ 9 & \frac{8}{3} & 5 \\ \frac{1}{2} - 5 \cdot \frac{2}{2} & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.14

Combine $- 5$ and $\frac{2}{2}$ .

$[\begin{array}{c} - 2 & \frac{37}{3} & - 4 \\ 9 & \frac{8}{3} & 5 \\ \frac{1}{2} + \frac{- 5 \cdot 2}{2} & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.15

Combine the numerators over the common denominator.

$[\begin{array}{c} - 2 & \frac{37}{3} & - 4 \\ 9 & \frac{8}{3} & 5 \\ \frac{1 - 5 \cdot 2}{2} & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.16

Simplify the numerator.

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Step 2.16.1

Multiply $- 5$ by $2$ .

$[\begin{array}{c} - 2 & \frac{37}{3} & - 4 \\ 9 & \frac{8}{3} & 5 \\ \frac{1 - 10}{2} & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.16.2

Subtract $10$ from $1$ .

$[\begin{array}{c} - 2 & \frac{37}{3} & - 4 \\ 9 & \frac{8}{3} & 5 \\ \frac{- 9}{2} & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.17

Move the negative in front of the fraction.

$[\begin{array}{c} - 2 & \frac{37}{3} & - 4 \\ 9 & \frac{8}{3} & 5 \\ - \frac{9}{2} & - 1 - 4 & 1 - 4 \end{array}]$

Step 2.18

Subtract $4$ from $- 1$ .

$[\begin{array}{c} - 2 & \frac{37}{3} & - 4 \\ 9 & \frac{8}{3} & 5 \\ - \frac{9}{2} & - 5 & 1 - 4 \end{array}]$

Step 2.19

Subtract $4$ from $1$ .

$[\begin{array}{c} - 2 & \frac{37}{3} & - 4 \\ 9 & \frac{8}{3} & 5 \\ - \frac{9}{2} & - 5 & - 3 \end{array}]$