Linear Algebra Examples

$[\begin{array}{c} 1 & 0 & 0 & 0 & - 8 \\ 0 & 2 & 0 & 0 & 0 \\ - 2 & 0 & 4 & 0 & 4 \\ 0 & - 12 & 0 & 8 & - 8 \end{array}]$

Step 1

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

$\sqrt{1^{2} + 0^{2} + 0^{2} + 0^{2} + {(- 8)}^{2} + 0^{2} + 2^{2} + 0^{2} + 0^{2} + 0^{2} + {(- 2)}^{2} + 0^{2} + 4^{2} + 0^{2} + 4^{2} + 0^{2} + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2

Simplify.

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Step 2.1

One to any power is one.

$\sqrt{1 + 0^{2} + 0^{2} + 0^{2} + {(- 8)}^{2} + 0^{2} + 2^{2} + 0^{2} + 0^{2} + 0^{2} + {(- 2)}^{2} + 0^{2} + 4^{2} + 0^{2} + 4^{2} + 0^{2} + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.2

Raising $0$ to any positive power yields $0$ .

$\sqrt{1 + 0 + 0^{2} + 0^{2} + {(- 8)}^{2} + 0^{2} + 2^{2} + 0^{2} + 0^{2} + 0^{2} + {(- 2)}^{2} + 0^{2} + 4^{2} + 0^{2} + 4^{2} + 0^{2} + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.3

Raising $0$ to any positive power yields $0$ .

$\sqrt{1 + 0 + 0 + 0^{2} + {(- 8)}^{2} + 0^{2} + 2^{2} + 0^{2} + 0^{2} + 0^{2} + {(- 2)}^{2} + 0^{2} + 4^{2} + 0^{2} + 4^{2} + 0^{2} + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.4

Raising $0$ to any positive power yields $0$ .

$\sqrt{1 + 0 + 0 + 0 + {(- 8)}^{2} + 0^{2} + 2^{2} + 0^{2} + 0^{2} + 0^{2} + {(- 2)}^{2} + 0^{2} + 4^{2} + 0^{2} + 4^{2} + 0^{2} + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.5

Raise $- 8$ to the power of $2$ .

$\sqrt{1 + 0 + 0 + 0 + 64 + 0^{2} + 2^{2} + 0^{2} + 0^{2} + 0^{2} + {(- 2)}^{2} + 0^{2} + 4^{2} + 0^{2} + 4^{2} + 0^{2} + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.6

Raising $0$ to any positive power yields $0$ .

$\sqrt{1 + 0 + 0 + 0 + 64 + 0 + 2^{2} + 0^{2} + 0^{2} + 0^{2} + {(- 2)}^{2} + 0^{2} + 4^{2} + 0^{2} + 4^{2} + 0^{2} + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.7

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

$\sqrt{1 + 0 + 0 + 0 + 64 + 0 + 4 + 0^{2} + 0^{2} + 0^{2} + {(- 2)}^{2} + 0^{2} + 4^{2} + 0^{2} + 4^{2} + 0^{2} + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.8

Raising $0$ to any positive power yields $0$ .

$\sqrt{1 + 0 + 0 + 0 + 64 + 0 + 4 + 0 + 0^{2} + 0^{2} + {(- 2)}^{2} + 0^{2} + 4^{2} + 0^{2} + 4^{2} + 0^{2} + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.9

Raising $0$ to any positive power yields $0$ .

$\sqrt{1 + 0 + 0 + 0 + 64 + 0 + 4 + 0 + 0 + 0^{2} + {(- 2)}^{2} + 0^{2} + 4^{2} + 0^{2} + 4^{2} + 0^{2} + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.10

Raising $0$ to any positive power yields $0$ .

$\sqrt{1 + 0 + 0 + 0 + 64 + 0 + 4 + 0 + 0 + 0 + {(- 2)}^{2} + 0^{2} + 4^{2} + 0^{2} + 4^{2} + 0^{2} + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.11

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

$\sqrt{1 + 0 + 0 + 0 + 64 + 0 + 4 + 0 + 0 + 0 + 4 + 0^{2} + 4^{2} + 0^{2} + 4^{2} + 0^{2} + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.12

Raising $0$ to any positive power yields $0$ .

$\sqrt{1 + 0 + 0 + 0 + 64 + 0 + 4 + 0 + 0 + 0 + 4 + 0 + 4^{2} + 0^{2} + 4^{2} + 0^{2} + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.13

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

$\sqrt{1 + 0 + 0 + 0 + 64 + 0 + 4 + 0 + 0 + 0 + 4 + 0 + 16 + 0^{2} + 4^{2} + 0^{2} + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.14

Raising $0$ to any positive power yields $0$ .

$\sqrt{1 + 0 + 0 + 0 + 64 + 0 + 4 + 0 + 0 + 0 + 4 + 0 + 16 + 0 + 4^{2} + 0^{2} + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.15

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

$\sqrt{1 + 0 + 0 + 0 + 64 + 0 + 4 + 0 + 0 + 0 + 4 + 0 + 16 + 0 + 16 + 0^{2} + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.16

Raising $0$ to any positive power yields $0$ .

$\sqrt{1 + 0 + 0 + 0 + 64 + 0 + 4 + 0 + 0 + 0 + 4 + 0 + 16 + 0 + 16 + 0 + {(- 12)}^{2} + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.17

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

$\sqrt{1 + 0 + 0 + 0 + 64 + 0 + 4 + 0 + 0 + 0 + 4 + 0 + 16 + 0 + 16 + 0 + 144 + 0^{2} + 8^{2} + {(- 8)}^{2}}$

Step 2.18

Raising $0$ to any positive power yields $0$ .

$\sqrt{1 + 0 + 0 + 0 + 64 + 0 + 4 + 0 + 0 + 0 + 4 + 0 + 16 + 0 + 16 + 0 + 144 + 0 + 8^{2} + {(- 8)}^{2}}$

Step 2.19

Raise $8$ to the power of $2$ .

$\sqrt{1 + 0 + 0 + 0 + 64 + 0 + 4 + 0 + 0 + 0 + 4 + 0 + 16 + 0 + 16 + 0 + 144 + 0 + 64 + {(- 8)}^{2}}$

Step 2.20

Raise $- 8$ to the power of $2$ .

$\sqrt{1 + 0 + 0 + 0 + 64 + 0 + 4 + 0 + 0 + 0 + 4 + 0 + 16 + 0 + 16 + 0 + 144 + 0 + 64 + 64}$

Step 2.21

Add $1$ and $0$ .

$\sqrt{1 + 0 + 0 + 64 + 0 + 4 + 0 + 0 + 0 + 4 + 0 + 16 + 0 + 16 + 0 + 144 + 0 + 64 + 64}$

Step 2.22

Add $1$ and $0$ .

$\sqrt{1 + 0 + 64 + 0 + 4 + 0 + 0 + 0 + 4 + 0 + 16 + 0 + 16 + 0 + 144 + 0 + 64 + 64}$

Step 2.23

Add $1$ and $0$ .

$\sqrt{1 + 64 + 0 + 4 + 0 + 0 + 0 + 4 + 0 + 16 + 0 + 16 + 0 + 144 + 0 + 64 + 64}$

Step 2.24

Add $1$ and $64$ .

$\sqrt{65 + 0 + 4 + 0 + 0 + 0 + 4 + 0 + 16 + 0 + 16 + 0 + 144 + 0 + 64 + 64}$

Step 2.25

Add $65$ and $0$ .

$\sqrt{65 + 4 + 0 + 0 + 0 + 4 + 0 + 16 + 0 + 16 + 0 + 144 + 0 + 64 + 64}$

Step 2.26

Add $65$ and $4$ .

$\sqrt{69 + 0 + 0 + 0 + 4 + 0 + 16 + 0 + 16 + 0 + 144 + 0 + 64 + 64}$

Step 2.27

Add $69$ and $0$ .

$\sqrt{69 + 0 + 0 + 4 + 0 + 16 + 0 + 16 + 0 + 144 + 0 + 64 + 64}$

Step 2.28

Add $69$ and $0$ .

$\sqrt{69 + 0 + 4 + 0 + 16 + 0 + 16 + 0 + 144 + 0 + 64 + 64}$

Step 2.29

Add $69$ and $0$ .

$\sqrt{69 + 4 + 0 + 16 + 0 + 16 + 0 + 144 + 0 + 64 + 64}$

Step 2.30

Add $69$ and $4$ .

$\sqrt{73 + 0 + 16 + 0 + 16 + 0 + 144 + 0 + 64 + 64}$

Step 2.31

Add $73$ and $0$ .

$\sqrt{73 + 16 + 0 + 16 + 0 + 144 + 0 + 64 + 64}$

Step 2.32

Add $73$ and $16$ .

$\sqrt{89 + 0 + 16 + 0 + 144 + 0 + 64 + 64}$

Step 2.33

Add $89$ and $0$ .

$\sqrt{89 + 16 + 0 + 144 + 0 + 64 + 64}$

Step 2.34

Add $89$ and $16$ .

$\sqrt{105 + 0 + 144 + 0 + 64 + 64}$

Step 2.35

Add $105$ and $0$ .

$\sqrt{105 + 144 + 0 + 64 + 64}$

Step 2.36

Add $105$ and $144$ .

$\sqrt{249 + 0 + 64 + 64}$

Step 2.37

Add $249$ and $0$ .

$\sqrt{249 + 64 + 64}$

Step 2.38

Add $249$ and $64$ .

$\sqrt{313 + 64}$

Step 2.39

Add $313$ and $64$ .

$\sqrt{377}$

$\sqrt{377}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{377}$

Decimal Form:

$19.41648783 \dots$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$