Linear Algebra Examples

$[\begin{array}{c} 1 & - 1 & - 1 & - 5 \\ - 3 & 2 & 4 & - 1 \\ - 10 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{array}]$

Step 1

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

$\sqrt{1^{2} + {(- 1)}^{2} + {(- 1)}^{2} + {(- 5)}^{2} + {(- 3)}^{2} + 2^{2} + 4^{2} + {(- 1)}^{2} + {(- 10)}^{2} + 10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2} + 15^{2} + 16^{2}}$

Step 2

Simplify.

Tap for more steps...

Step 2.1

One to any power is one.

$\sqrt{1 + {(- 1)}^{2} + {(- 1)}^{2} + {(- 5)}^{2} + {(- 3)}^{2} + 2^{2} + 4^{2} + {(- 1)}^{2} + {(- 10)}^{2} + 10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2} + 15^{2} + 16^{2}}$

Step 2.2

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

$\sqrt{1 + 1 + {(- 1)}^{2} + {(- 5)}^{2} + {(- 3)}^{2} + 2^{2} + 4^{2} + {(- 1)}^{2} + {(- 10)}^{2} + 10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2} + 15^{2} + 16^{2}}$

Step 2.3

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

$\sqrt{1 + 1 + 1 + {(- 5)}^{2} + {(- 3)}^{2} + 2^{2} + 4^{2} + {(- 1)}^{2} + {(- 10)}^{2} + 10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2} + 15^{2} + 16^{2}}$

Step 2.4

Raise $- 5$ to the power of $2$ .

$\sqrt{1 + 1 + 1 + 25 + {(- 3)}^{2} + 2^{2} + 4^{2} + {(- 1)}^{2} + {(- 10)}^{2} + 10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2} + 15^{2} + 16^{2}}$

Step 2.5

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

$\sqrt{1 + 1 + 1 + 25 + 9 + 2^{2} + 4^{2} + {(- 1)}^{2} + {(- 10)}^{2} + 10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2} + 15^{2} + 16^{2}}$

Step 2.6

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

$\sqrt{1 + 1 + 1 + 25 + 9 + 4 + 4^{2} + {(- 1)}^{2} + {(- 10)}^{2} + 10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2} + 15^{2} + 16^{2}}$

Step 2.7

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

$\sqrt{1 + 1 + 1 + 25 + 9 + 4 + 16 + {(- 1)}^{2} + {(- 10)}^{2} + 10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2} + 15^{2} + 16^{2}}$

Step 2.8

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

$\sqrt{1 + 1 + 1 + 25 + 9 + 4 + 16 + 1 + {(- 10)}^{2} + 10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2} + 15^{2} + 16^{2}}$

Step 2.9

Raise $- 10$ to the power of $2$ .

$\sqrt{1 + 1 + 1 + 25 + 9 + 4 + 16 + 1 + 100 + 10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2} + 15^{2} + 16^{2}}$

Step 2.10

Raise $10$ to the power of $2$ .

$\sqrt{1 + 1 + 1 + 25 + 9 + 4 + 16 + 1 + 100 + 100 + 11^{2} + 12^{2} + 13^{2} + 14^{2} + 15^{2} + 16^{2}}$

Step 2.11

Raise $11$ to the power of $2$ .

$\sqrt{1 + 1 + 1 + 25 + 9 + 4 + 16 + 1 + 100 + 100 + 121 + 12^{2} + 13^{2} + 14^{2} + 15^{2} + 16^{2}}$

Step 2.12

Raise $12$ to the power of $2$ .

$\sqrt{1 + 1 + 1 + 25 + 9 + 4 + 16 + 1 + 100 + 100 + 121 + 144 + 13^{2} + 14^{2} + 15^{2} + 16^{2}}$

Step 2.13

Raise $13$ to the power of $2$ .

$\sqrt{1 + 1 + 1 + 25 + 9 + 4 + 16 + 1 + 100 + 100 + 121 + 144 + 169 + 14^{2} + 15^{2} + 16^{2}}$

Step 2.14

Raise $14$ to the power of $2$ .

$\sqrt{1 + 1 + 1 + 25 + 9 + 4 + 16 + 1 + 100 + 100 + 121 + 144 + 169 + 196 + 15^{2} + 16^{2}}$

Step 2.15

Raise $15$ to the power of $2$ .

$\sqrt{1 + 1 + 1 + 25 + 9 + 4 + 16 + 1 + 100 + 100 + 121 + 144 + 169 + 196 + 225 + 16^{2}}$

Step 2.16

Raise $16$ to the power of $2$ .

$\sqrt{1 + 1 + 1 + 25 + 9 + 4 + 16 + 1 + 100 + 100 + 121 + 144 + 169 + 196 + 225 + 256}$

Step 2.17

Add $1$ and $1$ .

$\sqrt{2 + 1 + 25 + 9 + 4 + 16 + 1 + 100 + 100 + 121 + 144 + 169 + 196 + 225 + 256}$

Step 2.18

Add $2$ and $1$ .

$\sqrt{3 + 25 + 9 + 4 + 16 + 1 + 100 + 100 + 121 + 144 + 169 + 196 + 225 + 256}$

Step 2.19

Add $3$ and $25$ .

$\sqrt{28 + 9 + 4 + 16 + 1 + 100 + 100 + 121 + 144 + 169 + 196 + 225 + 256}$

Step 2.20

Add $28$ and $9$ .

$\sqrt{37 + 4 + 16 + 1 + 100 + 100 + 121 + 144 + 169 + 196 + 225 + 256}$

Step 2.21

Add $37$ and $4$ .

$\sqrt{41 + 16 + 1 + 100 + 100 + 121 + 144 + 169 + 196 + 225 + 256}$

Step 2.22

Add $41$ and $16$ .

$\sqrt{57 + 1 + 100 + 100 + 121 + 144 + 169 + 196 + 225 + 256}$

Step 2.23

Add $57$ and $1$ .

$\sqrt{58 + 100 + 100 + 121 + 144 + 169 + 196 + 225 + 256}$

Step 2.24

Add $58$ and $100$ .

$\sqrt{158 + 100 + 121 + 144 + 169 + 196 + 225 + 256}$

Step 2.25

Add $158$ and $100$ .

$\sqrt{258 + 121 + 144 + 169 + 196 + 225 + 256}$

Step 2.26

Add $258$ and $121$ .

$\sqrt{379 + 144 + 169 + 196 + 225 + 256}$

Step 2.27

Add $379$ and $144$ .

$\sqrt{523 + 169 + 196 + 225 + 256}$

Step 2.28

Add $523$ and $169$ .

$\sqrt{692 + 196 + 225 + 256}$

Step 2.29

Add $692$ and $196$ .

$\sqrt{888 + 225 + 256}$

Step 2.30

Add $888$ and $225$ .

$\sqrt{1113 + 256}$

Step 2.31

Add $1113$ and $256$ .

$\sqrt{1369}$

Step 2.32

Rewrite $1369$ as $37^{2}$ .

$\sqrt{37^{2}}$

Step 2.33

Pull terms out from under the radical, assuming positive real numbers.

$37$