Linear Algebra Examples

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

Step 1

The norm is the square root of the sum of squares of each element in the matrix.

$\sqrt{4^{2} + 1^{2} + {(- 2)}^{2} + 3^{2} + 1^{2} + {(\frac{1}{4})}^{2} + {(- \frac{1}{2})}^{2} + {(\frac{3}{4})}^{2} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2

Simplify.

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Step 2.1

Raise $4$ to the power of $2$ .

$\sqrt{16 + 1^{2} + {(- 2)}^{2} + 3^{2} + 1^{2} + {(\frac{1}{4})}^{2} + {(- \frac{1}{2})}^{2} + {(\frac{3}{4})}^{2} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.2

One to any power is one.

$\sqrt{16 + 1 + {(- 2)}^{2} + 3^{2} + 1^{2} + {(\frac{1}{4})}^{2} + {(- \frac{1}{2})}^{2} + {(\frac{3}{4})}^{2} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.3

Raise $- 2$ to the power of $2$ .

$\sqrt{16 + 1 + 4 + 3^{2} + 1^{2} + {(\frac{1}{4})}^{2} + {(- \frac{1}{2})}^{2} + {(\frac{3}{4})}^{2} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.4

Raise $3$ to the power of $2$ .

$\sqrt{16 + 1 + 4 + 9 + 1^{2} + {(\frac{1}{4})}^{2} + {(- \frac{1}{2})}^{2} + {(\frac{3}{4})}^{2} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.5

One to any power is one.

$\sqrt{16 + 1 + 4 + 9 + 1 + {(\frac{1}{4})}^{2} + {(- \frac{1}{2})}^{2} + {(\frac{3}{4})}^{2} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.6

Apply the product rule to $\frac{1}{4}$ .

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1^{2}}{4^{2}} + {(- \frac{1}{2})}^{2} + {(\frac{3}{4})}^{2} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.7

One to any power is one.

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{4^{2}} + {(- \frac{1}{2})}^{2} + {(\frac{3}{4})}^{2} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.8

Raise $4$ to the power of $2$ .

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{16} + {(- \frac{1}{2})}^{2} + {(\frac{3}{4})}^{2} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.9

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.9.1

Apply the product rule to $- \frac{1}{2}$ .

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{16} + {(- 1)}^{2} {(\frac{1}{2})}^{2} + {(\frac{3}{4})}^{2} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.9.2

Apply the product rule to $\frac{1}{2}$ .

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{16} + {(- 1)}^{2} \frac{1^{2}}{2^{2}} + {(\frac{3}{4})}^{2} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.10

Raise $- 1$ to the power of $2$ .

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{16} + 1 \frac{1^{2}}{2^{2}} + {(\frac{3}{4})}^{2} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.11

Multiply $\frac{1^{2}}{2^{2}}$ by $1$ .

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{16} + \frac{1^{2}}{2^{2}} + {(\frac{3}{4})}^{2} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.12

One to any power is one.

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{16} + \frac{1}{2^{2}} + {(\frac{3}{4})}^{2} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.13

Raise $2$ to the power of $2$ .

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{16} + \frac{1}{4} + {(\frac{3}{4})}^{2} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.14

Apply the product rule to $\frac{3}{4}$ .

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{16} + \frac{1}{4} + \frac{3^{2}}{4^{2}} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.15

Raise $3$ to the power of $2$ .

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{16} + \frac{1}{4} + \frac{9}{4^{2}} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.16

Raise $4$ to the power of $2$ .

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{16} + \frac{1}{4} + \frac{9}{16} + 2^{2} + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.17

Raise $2$ to the power of $2$ .

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{16} + \frac{1}{4} + \frac{9}{16} + 4 + {(\frac{1}{2})}^{2} + {(- 1)}^{2} + 1^{2}}$

Step 2.18

Apply the product rule to $\frac{1}{2}$ .

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{16} + \frac{1}{4} + \frac{9}{16} + 4 + \frac{1^{2}}{2^{2}} + {(- 1)}^{2} + 1^{2}}$

Step 2.19

One to any power is one.

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{16} + \frac{1}{4} + \frac{9}{16} + 4 + \frac{1}{2^{2}} + {(- 1)}^{2} + 1^{2}}$

Step 2.20

Raise $2$ to the power of $2$ .

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{16} + \frac{1}{4} + \frac{9}{16} + 4 + \frac{1}{4} + {(- 1)}^{2} + 1^{2}}$

Step 2.21

Raise $- 1$ to the power of $2$ .

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{16} + \frac{1}{4} + \frac{9}{16} + 4 + \frac{1}{4} + 1 + 1^{2}}$

Step 2.22

One to any power is one.

$\sqrt{16 + 1 + 4 + 9 + 1 + \frac{1}{16} + \frac{1}{4} + \frac{9}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.23

Add $16$ and $1$ .

$\sqrt{17 + 4 + 9 + 1 + \frac{1}{16} + \frac{1}{4} + \frac{9}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.24

Add $17$ and $4$ .

$\sqrt{21 + 9 + 1 + \frac{1}{16} + \frac{1}{4} + \frac{9}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.25

Add $21$ and $9$ .

$\sqrt{30 + 1 + \frac{1}{16} + \frac{1}{4} + \frac{9}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.26

Add $30$ and $1$ .

$\sqrt{31 + \frac{1}{16} + \frac{1}{4} + \frac{9}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.27

To write $31$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\sqrt{31 \cdot \frac{16}{16} + \frac{1}{16} + \frac{1}{4} + \frac{9}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.28

Combine $31$ and $\frac{16}{16}$ .

$\sqrt{\frac{31 \cdot 16}{16} + \frac{1}{16} + \frac{1}{4} + \frac{9}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.29

Combine the numerators over the common denominator.

$\sqrt{\frac{31 \cdot 16 + 1}{16} + \frac{1}{4} + \frac{9}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.30

Simplify the numerator.

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Step 2.30.1

Multiply $31$ by $16$ .

$\sqrt{\frac{496 + 1}{16} + \frac{1}{4} + \frac{9}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.30.2

Add $496$ and $1$ .

$\sqrt{\frac{497}{16} + \frac{1}{4} + \frac{9}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.31

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{497}{16} + \frac{1}{4} \cdot \frac{4}{4} + \frac{9}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.32

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.32.1

Multiply $\frac{1}{4}$ by $\frac{4}{4}$ .

$\sqrt{\frac{497}{16} + \frac{4}{4 \cdot 4} + \frac{9}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.32.2

Multiply $4$ by $4$ .

$\sqrt{\frac{497}{16} + \frac{4}{16} + \frac{9}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.33

Combine the numerators over the common denominator.

$\sqrt{\frac{497 + 4}{16} + \frac{9}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.34

Add $497$ and $4$ .

$\sqrt{\frac{501}{16} + \frac{9}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.35

Combine the numerators over the common denominator.

$\sqrt{\frac{501 + 9}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.36

Add $501$ and $9$ .

$\sqrt{\frac{510}{16} + 4 + \frac{1}{4} + 1 + 1}$

Step 2.37

To write $4$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\sqrt{\frac{510}{16} + 4 \cdot \frac{16}{16} + \frac{1}{4} + 1 + 1}$

Step 2.38

Combine $4$ and $\frac{16}{16}$ .

$\sqrt{\frac{510}{16} + \frac{4 \cdot 16}{16} + \frac{1}{4} + 1 + 1}$

Step 2.39

Combine the numerators over the common denominator.

$\sqrt{\frac{510 + 4 \cdot 16}{16} + \frac{1}{4} + 1 + 1}$

Step 2.40

Simplify the numerator.

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Step 2.40.1

Multiply $4$ by $16$ .

$\sqrt{\frac{510 + 64}{16} + \frac{1}{4} + 1 + 1}$

Step 2.40.2

Add $510$ and $64$ .

$\sqrt{\frac{574}{16} + \frac{1}{4} + 1 + 1}$

Step 2.41

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{574}{16} + \frac{1}{4} \cdot \frac{4}{4} + 1 + 1}$

Step 2.42

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.42.1

Multiply $\frac{1}{4}$ by $\frac{4}{4}$ .

$\sqrt{\frac{574}{16} + \frac{4}{4 \cdot 4} + 1 + 1}$

Step 2.42.2

Multiply $4$ by $4$ .

$\sqrt{\frac{574}{16} + \frac{4}{16} + 1 + 1}$

Step 2.43

Combine the numerators over the common denominator.

$\sqrt{\frac{574 + 4}{16} + 1 + 1}$

Step 2.44

Add $574$ and $4$ .

$\sqrt{\frac{578}{16} + 1 + 1}$

Step 2.45

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

$\sqrt{\frac{578}{16} + \frac{16}{16} + 1}$

Step 2.46

Combine the numerators over the common denominator.

$\sqrt{\frac{578 + 16}{16} + 1}$

Step 2.47

Add $578$ and $16$ .

$\sqrt{\frac{594}{16} + 1}$

Step 2.48

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

$\sqrt{\frac{594}{16} + \frac{16}{16}}$

Step 2.49

Combine the numerators over the common denominator.

$\sqrt{\frac{594 + 16}{16}}$

Step 2.50

Add $594$ and $16$ .

$\sqrt{\frac{610}{16}}$

Step 2.51

Cancel the common factor of $610$ and $16$ .

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Step 2.51.1

Factor $2$ out of $610$ .

$\sqrt{\frac{2 (305)}{16}}$

Step 2.51.2

Cancel the common factors.

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Step 2.51.2.1

Factor $2$ out of $16$ .

$\sqrt{\frac{2 \cdot 305}{2 \cdot 8}}$

Step 2.51.2.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 305}{2 \cdot 8}}$

Step 2.51.2.3

Rewrite the expression.

$\sqrt{\frac{305}{8}}$

Step 2.52

Rewrite $\sqrt{\frac{305}{8}}$ as $\frac{\sqrt{305}}{\sqrt{8}}$ .

$\frac{\sqrt{305}}{\sqrt{8}}$

Step 2.53

Simplify the denominator.

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Step 2.53.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.53.1.1

Factor $4$ out of $8$ .

$\frac{\sqrt{305}}{\sqrt{4 (2)}}$

Step 2.53.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{305}}{\sqrt{2^{2} \cdot 2}}$

Step 2.53.2

Pull terms out from under the radical.

$\frac{\sqrt{305}}{2 \sqrt{2}}$

Step 2.54

Multiply $\frac{\sqrt{305}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{305}}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.55

Combine and simplify the denominator.

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Step 2.55.1

Multiply $\frac{\sqrt{305}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{305} \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 2.55.2

Move $\sqrt{2}$ .

$\frac{\sqrt{305} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 2.55.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{305} \sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 2.55.4

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{305} \sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 2.55.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{305} \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 2.55.6

Add $1$ and $1$ .

$\frac{\sqrt{305} \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 2.55.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 2.55.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{\sqrt{305} \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 2.55.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{305} \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 2.55.7.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{305} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 2.55.7.4

Cancel the common factor of $2$ .

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Step 2.55.7.4.1

Cancel the common factor.

$\frac{\sqrt{305} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 2.55.7.4.2

Rewrite the expression.

$\frac{\sqrt{305} \sqrt{2}}{2 \cdot 2^{1}}$

Step 2.55.7.5

Evaluate the exponent.

$\frac{\sqrt{305} \sqrt{2}}{2 \cdot 2}$

Step 2.56

Simplify the numerator.

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Step 2.56.1

Combine using the product rule for radicals.

$\frac{\sqrt{305 \cdot 2}}{2 \cdot 2}$

Step 2.56.2

Multiply $305$ by $2$ .

$\frac{\sqrt{610}}{2 \cdot 2}$

Step 2.57

Multiply $2$ by $2$ .

$\frac{\sqrt{610}}{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{610}}{4}$

Decimal Form:

$6.17454451 \dots$