Linear Algebra Examples

$[\begin{array}{c} 1 & 2 & - 9 & - 8 \\ 0 & 1 & 7 & - 3 \\ 0 & 0 & 1 & 7 \end{array}]$

Step 1

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

$\sqrt{1^{2} + 2^{2} + {(- 9)}^{2} + {(- 8)}^{2} + 0^{2} + 1^{2} + 7^{2} + {(- 3)}^{2} + 0^{2} + 0^{2} + 1^{2} + 7^{2}}$

Step 2

Simplify.

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Step 2.1

One to any power is one.

$\sqrt{1 + 2^{2} + {(- 9)}^{2} + {(- 8)}^{2} + 0^{2} + 1^{2} + 7^{2} + {(- 3)}^{2} + 0^{2} + 0^{2} + 1^{2} + 7^{2}}$

Step 2.2

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

$\sqrt{1 + 4 + {(- 9)}^{2} + {(- 8)}^{2} + 0^{2} + 1^{2} + 7^{2} + {(- 3)}^{2} + 0^{2} + 0^{2} + 1^{2} + 7^{2}}$

Step 2.3

Raise $- 9$ to the power of $2$ .

$\sqrt{1 + 4 + 81 + {(- 8)}^{2} + 0^{2} + 1^{2} + 7^{2} + {(- 3)}^{2} + 0^{2} + 0^{2} + 1^{2} + 7^{2}}$

Step 2.4

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

$\sqrt{1 + 4 + 81 + 64 + 0^{2} + 1^{2} + 7^{2} + {(- 3)}^{2} + 0^{2} + 0^{2} + 1^{2} + 7^{2}}$

Step 2.5

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

$\sqrt{1 + 4 + 81 + 64 + 0 + 1^{2} + 7^{2} + {(- 3)}^{2} + 0^{2} + 0^{2} + 1^{2} + 7^{2}}$

Step 2.6

One to any power is one.

$\sqrt{1 + 4 + 81 + 64 + 0 + 1 + 7^{2} + {(- 3)}^{2} + 0^{2} + 0^{2} + 1^{2} + 7^{2}}$

Step 2.7

Raise $7$ to the power of $2$ .

$\sqrt{1 + 4 + 81 + 64 + 0 + 1 + 49 + {(- 3)}^{2} + 0^{2} + 0^{2} + 1^{2} + 7^{2}}$

Step 2.8

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

$\sqrt{1 + 4 + 81 + 64 + 0 + 1 + 49 + 9 + 0^{2} + 0^{2} + 1^{2} + 7^{2}}$

Step 2.9

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

$\sqrt{1 + 4 + 81 + 64 + 0 + 1 + 49 + 9 + 0 + 0^{2} + 1^{2} + 7^{2}}$

Step 2.10

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

$\sqrt{1 + 4 + 81 + 64 + 0 + 1 + 49 + 9 + 0 + 0 + 1^{2} + 7^{2}}$

Step 2.11

One to any power is one.

$\sqrt{1 + 4 + 81 + 64 + 0 + 1 + 49 + 9 + 0 + 0 + 1 + 7^{2}}$

Step 2.12

Raise $7$ to the power of $2$ .

$\sqrt{1 + 4 + 81 + 64 + 0 + 1 + 49 + 9 + 0 + 0 + 1 + 49}$

Step 2.13

Add $1$ and $4$ .

$\sqrt{5 + 81 + 64 + 0 + 1 + 49 + 9 + 0 + 0 + 1 + 49}$

Step 2.14

Add $5$ and $81$ .

$\sqrt{86 + 64 + 0 + 1 + 49 + 9 + 0 + 0 + 1 + 49}$

Step 2.15

Add $86$ and $64$ .

$\sqrt{150 + 0 + 1 + 49 + 9 + 0 + 0 + 1 + 49}$

Step 2.16

Add $150$ and $0$ .

$\sqrt{150 + 1 + 49 + 9 + 0 + 0 + 1 + 49}$

Step 2.17

Add $150$ and $1$ .

$\sqrt{151 + 49 + 9 + 0 + 0 + 1 + 49}$

Step 2.18

Add $151$ and $49$ .

$\sqrt{200 + 9 + 0 + 0 + 1 + 49}$

Step 2.19

Add $200$ and $9$ .

$\sqrt{209 + 0 + 0 + 1 + 49}$

Step 2.20

Add $209$ and $0$ .

$\sqrt{209 + 0 + 1 + 49}$

Step 2.21

Add $209$ and $0$ .

$\sqrt{209 + 1 + 49}$

Step 2.22

Add $209$ and $1$ .

$\sqrt{210 + 49}$

Step 2.23

Add $210$ and $49$ .

$\sqrt{259}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{259}$

Decimal Form:

$16.09347693 \dots$