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
}
\pagebreak
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
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
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
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 &
}