Linear Algebra Examples

$[\begin{array}{c} 4 & 5 & - 5 & - 3 \\ 2 & 7 & 4 & - 3 \end{array}]$

Step 1

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

$\sqrt{4^{2} + 5^{2} + {(- 5)}^{2} + {(- 3)}^{2} + 2^{2} + 7^{2} + 4^{2} + {(- 3)}^{2}}$

Step 2

Simplify.

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Step 2.1

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

$\sqrt{16 + 5^{2} + {(- 5)}^{2} + {(- 3)}^{2} + 2^{2} + 7^{2} + 4^{2} + {(- 3)}^{2}}$

Step 2.2

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

$\sqrt{16 + 25 + {(- 5)}^{2} + {(- 3)}^{2} + 2^{2} + 7^{2} + 4^{2} + {(- 3)}^{2}}$

Step 2.3

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

$\sqrt{16 + 25 + 25 + {(- 3)}^{2} + 2^{2} + 7^{2} + 4^{2} + {(- 3)}^{2}}$

Step 2.4

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

$\sqrt{16 + 25 + 25 + 9 + 2^{2} + 7^{2} + 4^{2} + {(- 3)}^{2}}$

Step 2.5

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

$\sqrt{16 + 25 + 25 + 9 + 4 + 7^{2} + 4^{2} + {(- 3)}^{2}}$

Step 2.6

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

$\sqrt{16 + 25 + 25 + 9 + 4 + 49 + 4^{2} + {(- 3)}^{2}}$

Step 2.7

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

$\sqrt{16 + 25 + 25 + 9 + 4 + 49 + 16 + {(- 3)}^{2}}$

Step 2.8

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

$\sqrt{16 + 25 + 25 + 9 + 4 + 49 + 16 + 9}$

Step 2.9

Add $16$ and $25$ .

$\sqrt{41 + 25 + 9 + 4 + 49 + 16 + 9}$

Step 2.10

Add $41$ and $25$ .

$\sqrt{66 + 9 + 4 + 49 + 16 + 9}$

Step 2.11

Add $66$ and $9$ .

$\sqrt{75 + 4 + 49 + 16 + 9}$

Step 2.12

Add $75$ and $4$ .

$\sqrt{79 + 49 + 16 + 9}$

Step 2.13

Add $79$ and $49$ .

$\sqrt{128 + 16 + 9}$

Step 2.14

Add $128$ and $16$ .

$\sqrt{144 + 9}$

Step 2.15

Add $144$ and $9$ .

$\sqrt{153}$

Step 2.16

Rewrite $153$ as $3^{2} \cdot 17$ .

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Step 2.16.1

Factor $9$ out of $153$ .

$\sqrt{9 (17)}$

Step 2.16.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2} \cdot 17}$

Step 2.17

Pull terms out from under the radical.

$3 \sqrt{17}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{17}$

Decimal Form:

$12.36931687 \dots$