Linear Algebra Examples

$[\begin{array}{c} 1 & 4 & 7 \\ 5 & 9 & - 9 \\ 0 & 7 & 9 \end{array}] - [\begin{array}{c} \frac{46}{5} & - \frac{12}{5} & \frac{3}{5} \\ \frac{12}{5} & - \frac{3}{5} & \frac{1}{5} \\ - \frac{3}{5} & \frac{1}{5} & 0 \end{array}]$

Step 1

Subtract the corresponding elements.

$[\begin{array}{c} 1 - \frac{46}{5} & 4 + \frac{12}{5} & 7 - \frac{3}{5} \\ 5 - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2

Simplify each element.

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Step 2.1

Write $1$ as a fraction with a common denominator.

$[\begin{array}{c} \frac{5}{5} - \frac{46}{5} & 4 + \frac{12}{5} & 7 - \frac{3}{5} \\ 5 - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.2

Combine the numerators over the common denominator.

$[\begin{array}{c} \frac{5 - 46}{5} & 4 + \frac{12}{5} & 7 - \frac{3}{5} \\ 5 - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.3

Subtract $46$ from $5$ .

$[\begin{array}{c} \frac{- 41}{5} & 4 + \frac{12}{5} & 7 - \frac{3}{5} \\ 5 - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.4

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{41}{5} & 4 + \frac{12}{5} & 7 - \frac{3}{5} \\ 5 - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.5

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$[\begin{array}{c} - \frac{41}{5} & 4 \cdot \frac{5}{5} + \frac{12}{5} & 7 - \frac{3}{5} \\ 5 - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.6

Combine $4$ and $\frac{5}{5}$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{4 \cdot 5}{5} + \frac{12}{5} & 7 - \frac{3}{5} \\ 5 - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.7

Combine the numerators over the common denominator.

$[\begin{array}{c} - \frac{41}{5} & \frac{4 \cdot 5 + 12}{5} & 7 - \frac{3}{5} \\ 5 - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Multiply $4$ by $5$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{20 + 12}{5} & 7 - \frac{3}{5} \\ 5 - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.8.2

Add $20$ and $12$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & 7 - \frac{3}{5} \\ 5 - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.9

To write $7$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & 7 \cdot \frac{5}{5} - \frac{3}{5} \\ 5 - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.10

Combine $7$ and $\frac{5}{5}$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{7 \cdot 5}{5} - \frac{3}{5} \\ 5 - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.11

Combine the numerators over the common denominator.

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{7 \cdot 5 - 3}{5} \\ 5 - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.12

Simplify the numerator.

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Step 2.12.1

Multiply $7$ by $5$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{35 - 3}{5} \\ 5 - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.12.2

Subtract $3$ from $35$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ 5 - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.13

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

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ 5 \cdot \frac{5}{5} - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.14

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

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{5 \cdot 5}{5} - \frac{12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.15

Combine the numerators over the common denominator.

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{5 \cdot 5 - 12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.16

Simplify the numerator.

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Step 2.16.1

Multiply $5$ by $5$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{25 - 12}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.16.2

Subtract $12$ from $25$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & 9 + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.17

To write $9$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & 9 \cdot \frac{5}{5} + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.18

Combine $9$ and $\frac{5}{5}$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{9 \cdot 5}{5} + \frac{3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.19

Combine the numerators over the common denominator.

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{9 \cdot 5 + 3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.20

Simplify the numerator.

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Step 2.20.1

Multiply $9$ by $5$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{45 + 3}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.20.2

Add $45$ and $3$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{48}{5} & - 9 - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.21

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{48}{5} & - 9 \cdot \frac{5}{5} - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.22

Combine $- 9$ and $\frac{5}{5}$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{48}{5} & \frac{- 9 \cdot 5}{5} - \frac{1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.23

Combine the numerators over the common denominator.

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{48}{5} & \frac{- 9 \cdot 5 - 1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.24

Simplify the numerator.

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Step 2.24.1

Multiply $- 9$ by $5$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{48}{5} & \frac{- 45 - 1}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.24.2

Subtract $1$ from $- 45$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{48}{5} & \frac{- 46}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.25

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{48}{5} & - \frac{46}{5} \\ 0 + \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.26

Add $0$ and $\frac{3}{5}$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{48}{5} & - \frac{46}{5} \\ \frac{3}{5} & 7 - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.27

To write $7$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{48}{5} & - \frac{46}{5} \\ \frac{3}{5} & 7 \cdot \frac{5}{5} - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.28

Combine $7$ and $\frac{5}{5}$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{48}{5} & - \frac{46}{5} \\ \frac{3}{5} & \frac{7 \cdot 5}{5} - \frac{1}{5} & 9 - 0 \end{array}]$

Step 2.29

Combine the numerators over the common denominator.

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{48}{5} & - \frac{46}{5} \\ \frac{3}{5} & \frac{7 \cdot 5 - 1}{5} & 9 - 0 \end{array}]$

Step 2.30

Simplify the numerator.

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Step 2.30.1

Multiply $7$ by $5$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{48}{5} & - \frac{46}{5} \\ \frac{3}{5} & \frac{35 - 1}{5} & 9 - 0 \end{array}]$

Step 2.30.2

Subtract $1$ from $35$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{48}{5} & - \frac{46}{5} \\ \frac{3}{5} & \frac{34}{5} & 9 - 0 \end{array}]$

Step 2.31

Subtract $0$ from $9$ .

$[\begin{array}{c} - \frac{41}{5} & \frac{32}{5} & \frac{32}{5} \\ \frac{13}{5} & \frac{48}{5} & - \frac{46}{5} \\ \frac{3}{5} & \frac{34}{5} & 9 \end{array}]$