uni/mat1001/gr2.md

Innlevering 2

A2

1

A) \mmatrix{4&1} \mmatrix{3&0 \ 2&7} = \mmatrix{4 * 3 + 1 * 2 & 4 * 0 + 1 * 0} = \mmatrix{14 & 0}

B) \mmatrix{1&1 \ 1&1} \mmatrix{2&2 \ 2&2} = \mmatrix{1 * 2 + 1 * 2 & 1 * 2 + 1 * 2 \ 1 * 2 + 1 * 2 & 1 * 2 + 1 * 2} = \mmatrix{4 & 4 \ 4 & 4}

C) \mmatrix{2&3&1 \ 4&8&2} \mmatrix{5&1 \ 1&0 \ 3&2} = \mmatrix{2 * 5+3 * 1 + 1 * 3 & 2 * 1 + 3 * 0 + 1 * 2 \ 4 * 5 + 8 * 1 + 2 * 3 & 4 * 1 + 8 * 0 + 2 * 2} = \mmatrix{16 & 4 \ 34 & 8}

D) \mmatrix{4&5 \ 2&1 \ 1&-1 \ 3&7} \mmatrix{2&1&3&0 \ 4&1&2&1} = \mmatrix{

4 * 2 + 5 * 4    & 4 * 1 + 5 * 1    & 4 * 3 + 5 * 2    & 4 * 0 + 5 * 1    \\
2 * 2 + 1 * 4    & 2 * 1 + 1 * 1    & 2 * 3 + 1 * 2    & 2 * 0 + 1 * 1    \\
1 * 2 + (-1 * 4) & 1 * 1 + (-1 * 1) & 1 * 3 + (-1 * 2) & 1 * 0 + (-1 * 1) \\
3 * 2 + 7 * 4    & 3 * 1 + 7 * 1    & 3 * 3 + 7 * 3    & 3 * 0 + 7 * 1

} = \mmatrix{

28 & 9  & 22 & 5  \\
8  & 3  & 8  & 1  \\
-2 & 0  & 1  & -1 \\
34 & 10 & 30 & 7

}

2

• A = \mmatrix{4&0 \ 2&3 \ 6&1}
• B = \mmatrix{3&1 \ 7&2}
• C = \mmatrix{8&3&2 \ 5&0&1 \ 6&6&7}
• D = \mmatrix{0&1&5 \ 2&4&2}
• E = \mmatrix{9&1&2 \ 0&4&4 \ 5&0&7}

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A)

* A: 2x3
* B: 2x2
* C: 3x3
* D: 3x2
* E: 3x3

B)

1) AB: Udefinert, fordi 3 != 2
2) AB + C: Udefinert, fordi AB er udefinert
3) 3E: Definert, fordi skalarprodukt er alltid definert
4) DA - B: Udefinert, fordi DA er en 3x3-matrise og B er en 2x2-matrise
5) BD + A: Udefinert, fordi BD er udefinert
6) ABD + 2CE: Udefinert, fordi AB er udefinert

12

A) det \mmatrix{1&0 \ 0&1} = 1 * 1 - 0 * 0 = 1

B) det \mmatrix{15&2 \ 0&8} = 15 * 8 - 0 * 2 = 120

C) det \mmatrix{

2 & 0 & 0 \\
1 & 3 & 4 \\
1 & 4 & 2

} = 2 det \mmatrix{

3 & 4 \\
4 & 3

} - 0 det \mmatrix{

1 & 4 \\
4 & 2

} + 0 det \mmatrix{

3 & 4 \\
4 & 2

} =
2(3 * 3 - 4 * 4) - 0 + 0 = 2(9 * 16) = 2 * 144 = 288

D) det \mmatrix{30&2 \ -40&4} = 120 - (-80) = 200

13

A) det \mmatrix{3&5 \ -2&4} = 12 - (-10) = 22 B) det \mmatrix{-5&6 \ -7&-2} = 10 - (-42) = 52 C) det \mmatrix{

-2 & 1 & 4 \\
3 & 5 & -7 \\
1 & 6 & 2

} = -2 det \mmatrix{

5 & -7 \\
6 & 2

} - 1 \mmatrix{

3 & -7 \\
1 & 2

} + 4 \mmatrix{

3 & 5 \\
1 & 6

} =
-2(5 * 2 - 6 * -7) - 1(3 * 2 - 1 * -7) + 4(3 * 6 - 1 * 5) = 104 - 13 + 52 = 143

B1

5

• A = \mmatrix{ 2 & -3 & 1 \ -1 & 2 & 4 \ 1 & 3 & -1 }

1) A^2 = \mmatrix{2&-3&1 \ -1&2&4 \ 1&3&-1} \mmatrix{2&-3&1 \ -1&2&4 \ 1&3&-1} = \mmatrix{

2 * 2 + -3 * -1 + 1 * 1 & 

}

6

7