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}

\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

### 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