dictionary
ypperlig för att representera grannlistor.$\boldsymbol{A}^3 = \boldsymbol{A} \times \boldsymbol{A} \times \boldsymbol{A}= \begin{bmatrix} 0 & 0 & 1 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{bmatrix} $
Följande vägar av längden 3 finns i $G$
$ \boldsymbol{A} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$, $\qquad \boldsymbol{B} = \begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix} \qquad $
$$\begin{align} \notag\boldsymbol{A} \times \boldsymbol{B} \notag&= \phantom{\begin{bmatrix} A_{1,*}\cdot B_{*,1} & A_{1,*}\cdot B_{*,2} \\ A_{2,*}\cdot B_{*,1} & A_{2,*}\cdot B_{*,2} \end{bmatrix} =}\\ \notag&\phantom{=\begin{bmatrix} 1 \cdot 7 + 2 \cdot 9 + 3 \cdot 11 & 1 \cdot 8 + 2 \cdot 10 + 3 \cdot 12 \\ 4 \cdot 7 + 5 \cdot 9 + 6 \cdot 11 & 4 \cdot 8 + 5 \cdot 10 + 6 \cdot 12 \end{bmatrix} =}\\ \notag&\phantom{=\begin{bmatrix} 7 + 18 + 33 & 8 + 20 + 36 \\ 28 + 45 + 66 & 32 + 50 + 72 \end{bmatrix} =}\\ \notag&\phantom{=\begin{bmatrix} 58 & 64 \\ 139 & 154 \end{bmatrix}} \end{align}$$$ \boldsymbol{A} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$, $\qquad \boldsymbol{B} = \begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix} \qquad $
$$\begin{align} \notag\boldsymbol{A} \times \boldsymbol{B} \notag&= \begin{bmatrix} \boldsymbol{a}_{1,*}\cdot \boldsymbol{b}_{*,1} & \boldsymbol{a}_{1,*}\cdot \boldsymbol{b}_{*,2} \\ \boldsymbol{a}_{2,*}\cdot \boldsymbol{b}_{*,1} & \boldsymbol{a}_{2,*}\cdot \boldsymbol{b}_{*,2} \end{bmatrix} =\\ \notag&\phantom{=\begin{bmatrix} {\color{BurntOrange}1} \cdot {\color{ForestGreen}7} + {\color{BurntOrange}2} \cdot {\color{ForestGreen}9} + {\color{BurntOrange}3} \cdot {\color{ForestGreen}11} & 1 \cdot 8 + 2 \cdot 10 + 3 \cdot 12 \\ 4 \cdot 7 + 5 \cdot 9 + 6 \cdot 11 & 4 \cdot 8 + 5 \cdot 10 + 6 \cdot 12 \end{bmatrix} =}\\ \notag&\phantom{=\begin{bmatrix} 7 + 18 + 33 & 8 + 20 + 36 \\ 28 + 45 + 66 & 32 + 50 + 72 \end{bmatrix} =}\\ \notag&\phantom{=\begin{bmatrix} 58 & 64 \\ 139 & 154 \end{bmatrix}} \end{align}$$$ \require{color} \boldsymbol{A} = \begin{bmatrix} {\color{BurntOrange}1} & {\color{BurntOrange}2} & {\color{BurntOrange} 3} \\ 4 & 5 & 6 \end{bmatrix}$, $\qquad \boldsymbol{B} = \begin{bmatrix} {\color{ForestGreen}7} & 8 \\ {\color{ForestGreen}9} & 10 \\ {\color{ForestGreen}11} & 12 \end{bmatrix} \qquad $
$$\begin{align} \notag\boldsymbol{A} \times \boldsymbol{B} \notag&= \begin{bmatrix} \boldsymbol{a}_{1,*}\cdot \boldsymbol{b}_{*,1} & \boldsymbol{a}_{1,*}\cdot \boldsymbol{b}_{*,2} \\ \boldsymbol{a}_{2,*}\cdot \boldsymbol{b}_{*,1} & \boldsymbol{a}_{2,*}\cdot \boldsymbol{b}_{*,2} \end{bmatrix} =\\ \notag&=\begin{bmatrix} {\color{BurntOrange}1} \cdot {\color{ForestGreen}7} + {\color{BurntOrange}2} \cdot {\color{ForestGreen}9} + {\color{BurntOrange}3} \cdot {\color{ForestGreen}11} & \phantom{1 \cdot 8 + 2 \cdot 10 + 3 \cdot 12} \\ \phantom{4 \cdot 7 + 5 \cdot 9 + 6 \cdot 11} & \phantom{4 \cdot 8 + 5 \cdot 10 + 6 \cdot 12} \end{bmatrix} =\\ \notag&\phantom{=\begin{bmatrix} 7 + 18 + 33 & 8 + 20 + 36 \\ 28 + 45 + 66 & 32 + 50 + 72 \end{bmatrix} =}\\ \notag&\phantom{=\begin{bmatrix} 58 & 64 \\ 139 & 154 \end{bmatrix}} \end{align}$$$ \boldsymbol{A} = \begin{bmatrix} {\color{BurntOrange}1} & {\color{BurntOrange}2} & {\color{BurntOrange} 3} \\ 4 & 5 & 6 \end{bmatrix}$, $\qquad \boldsymbol{B} = \begin{bmatrix} 7 & {\color{ForestGreen}8} \\ 9 & {\color{ForestGreen}10} \\ 11 & {\color{ForestGreen}12} \end{bmatrix} \qquad $
$$\begin{align} \notag\boldsymbol{A} \times \boldsymbol{B} \notag&= \begin{bmatrix} \boldsymbol{a}_{1,*}\cdot \boldsymbol{b}_{*,1} & \boldsymbol{a}_{1,*}\cdot \boldsymbol{b}_{*,2} \\ \boldsymbol{a}_{2,*}\cdot \boldsymbol{b}_{*,1} & \boldsymbol{a}_{2,*}\cdot \boldsymbol{b}_{*,2} \end{bmatrix} =\\ \notag&=\begin{bmatrix} 1 \cdot 7 + 2 \cdot 9 + 3 \cdot 11 & {\color{BurntOrange}1} \cdot {\color{ForestGreen}8} + {\color{BurntOrange}2} \cdot {\color{ForestGreen}10} + {\color{BurntOrange}3} \cdot {\color{ForestGreen}12} \\ \phantom{4 \cdot 7 + 5 \cdot 9 + 6 \cdot 11} & \phantom{4 \cdot 8 + 5 \cdot 10 + 6 \cdot 12} \end{bmatrix} =\\ \notag&\phantom{=\begin{bmatrix} 7 + 18 + 33 & 8 + 20 + 36 \\ 28 + 45 + 66 & 32 + 50 + 72 \end{bmatrix} =}\\ \notag&\phantom{=\begin{bmatrix} 58 & 64 \\ 139 & 154 \end{bmatrix}} \end{align}$$$ \boldsymbol{A} = \begin{bmatrix} 1 & 2 & 3 \\ {\color{BurntOrange}4} & {\color{BurntOrange}5} & {\color{BurntOrange}6} \end{bmatrix}$, $\qquad \boldsymbol{B} = \begin{bmatrix} {\color{ForestGreen}7} & 8 \\ {\color{ForestGreen}9} & 10 \\ {\color{ForestGreen}11} & 12 \end{bmatrix} \qquad $
$$\begin{align} \notag\boldsymbol{A} \times \boldsymbol{B} \notag&= \begin{bmatrix} \boldsymbol{a}_{1,*}\cdot \boldsymbol{b}_{*,1} & \boldsymbol{a}_{1,*}\cdot \boldsymbol{b}_{*,2} \\ \boldsymbol{a}_{2,*}\cdot \boldsymbol{b}_{*,1} & \boldsymbol{a}_{2,*}\cdot \boldsymbol{b}_{*,2} \end{bmatrix} =\\ \notag&=\begin{bmatrix} 1 \cdot 7 + 2 \cdot 9 + 3 \cdot 11 & 1 \cdot 8 + 2 \cdot 10 + 3 \cdot 12 \\ {\color{BurntOrange}4} \cdot {\color{ForestGreen}7} + {\color{BurntOrange}5} \cdot {\color{ForestGreen}9} + {\color{BurntOrange}6} \cdot {\color{ForestGreen}11} & \phantom{4 \cdot 8 + 5 \cdot 10 + 6 \cdot 12} \end{bmatrix} =\\ \notag&\phantom{=\begin{bmatrix} 7 + 18 + 33 & 8 + 20 + 36 \\ 28 + 45 + 66 & 32 + 50 + 72 \end{bmatrix} =}\\ \notag&\phantom{=\begin{bmatrix} 58 & 64 \\ 139 & 154 \end{bmatrix}} \end{align}$$$ \boldsymbol{A} = \begin{bmatrix} 1 & 2 & 3 \\ {\color{BurntOrange}4} & {\color{BurntOrange}5} & {\color{BurntOrange}6} \end{bmatrix}$, $\qquad \boldsymbol{B} = \begin{bmatrix} 7 & {\color{ForestGreen}8} \\ 9 & {\color{ForestGreen}10} \\ 11 & {\color{ForestGreen}12} \end{bmatrix} \qquad $
$$\begin{align} \notag\boldsymbol{A} \times \boldsymbol{B} \notag&= \begin{bmatrix} \boldsymbol{a}_{1,*}\cdot \boldsymbol{b}_{*,1} & \boldsymbol{a}_{1,*}\cdot \boldsymbol{b}_{*,2} \\ \boldsymbol{a}_{2,*}\cdot \boldsymbol{b}_{*,1} & \boldsymbol{a}_{2,*}\cdot \boldsymbol{b}_{*,2} \end{bmatrix} =\\ \notag&=\begin{bmatrix} 1 \cdot 7 + 2 \cdot 9 + 3 \cdot 11 & 1 \cdot 8 + 2 \cdot 10 + 3 \cdot 12 \\ 4 \cdot 7 + 5 \cdot 9 + 6 \cdot 11 & {\color{BurntOrange}4} \cdot {\color{ForestGreen}8} + {\color{BurntOrange}5} \cdot {\color{ForestGreen}10} + {\color{BurntOrange}6} \cdot {\color{ForestGreen}12} \end{bmatrix} =\\ \notag&\phantom{=\begin{bmatrix} 7 + 18 + 33 & 8 + 20 + 36 \\ 28 + 45 + 66 & 32 + 50 + 72 \end{bmatrix} =}\\ \notag&\phantom{=\begin{bmatrix} 58 & 64 \\ 139 & 154 \end{bmatrix}} \end{align}$$$ \boldsymbol{A} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$, $\qquad \boldsymbol{B} = \begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix} \qquad $
$$\begin{align} \notag\boldsymbol{A} \times \boldsymbol{B} \notag&= \begin{bmatrix} \boldsymbol{a}_{1,*}\cdot \boldsymbol{b}_{*,1} & \boldsymbol{a}_{1,*}\cdot \boldsymbol{b}_{*,2} \\ \boldsymbol{a}_{2,*}\cdot \boldsymbol{b}_{*,1} & \boldsymbol{a}_{2,*}\cdot \boldsymbol{b}_{*,2} \end{bmatrix} =\\ \notag&=\begin{bmatrix} 1 \cdot 7 + 2 \cdot 9 + 3 \cdot 11 & 1 \cdot 8 + 2 \cdot 10 + 3 \cdot 12 \\ 4 \cdot 7 + 5 \cdot 9 + 6 \cdot 11 & 4 \cdot 8 + 5 \cdot 10 + 6 \cdot 12 \end{bmatrix} =\\ \notag&\phantom{=\begin{bmatrix} 7 + 18 + 33 & 8 + 20 + 36 \\ 28 + 45 + 66 & 32 + 50 + 72 \end{bmatrix} =}\\ \notag&\phantom{=\begin{bmatrix} 58 & 64 \\ 139 & 154 \end{bmatrix}} \end{align}$$$ \boldsymbol{A} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$, $\qquad \boldsymbol{B} = \begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix} \qquad $
$$\begin{align} \notag\boldsymbol{A} \times \boldsymbol{B} \notag&= \begin{bmatrix} \boldsymbol{a}_{1,*}\cdot \boldsymbol{b}_{*,1} & \boldsymbol{a}_{1,*}\cdot \boldsymbol{b}_{*,2} \\ \boldsymbol{a}_{2,*}\cdot \boldsymbol{b}_{*,1} & \boldsymbol{a}_{2,*}\cdot \boldsymbol{b}_{*,2} \end{bmatrix} =\\ \notag&=\begin{bmatrix} 1 \cdot 7 + 2 \cdot 9 + 3 \cdot 11 & 1 \cdot 8 + 2 \cdot 10 + 3 \cdot 12 \\ 4 \cdot 7 + 5 \cdot 9 + 6 \cdot 11 & 4 \cdot 8 + 5 \cdot 10 + 6 \cdot 12 \end{bmatrix} =\\ \notag&=\begin{bmatrix} 7 + 18 + 33 & 8 + 20 + 36 \\ 28 + 45 + 66 & 32 + 50 + 72 \end{bmatrix} =\\ \notag&\phantom{=\begin{bmatrix} 58 & 64 \\ 139 & 154 \end{bmatrix}} \end{align}$$$ \boldsymbol{A} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$, $\qquad \boldsymbol{B} = \begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix} \qquad $
$$\begin{align} \notag\boldsymbol{A} \times \boldsymbol{B} \notag&= \begin{bmatrix} \boldsymbol{a}_{1,*}\cdot \boldsymbol{b}_{*,1} & \boldsymbol{a}_{1,*}\cdot \boldsymbol{b}_{*,2} \\ \boldsymbol{a}_{2,*}\cdot \boldsymbol{b}_{*,1} & \boldsymbol{a}_{2,*}\cdot \boldsymbol{b}_{*,2} \end{bmatrix} =\\ \notag&=\begin{bmatrix} 1 \cdot 7 + 2 \cdot 9 + 3 \cdot 11 & 1 \cdot 8 + 2 \cdot 10 + 3 \cdot 12 \\ 4 \cdot 7 + 5 \cdot 9 + 6 \cdot 11 & 4 \cdot 8 + 5 \cdot 10 + 6 \cdot 12 \end{bmatrix} =\\ \notag&=\begin{bmatrix} 7 + 18 + 33 & 8 + 20 + 36 \\ 28 + 45 + 66 & 32 + 50 + 72 \end{bmatrix} =\\ \notag&=\begin{bmatrix} 58 & 64 \\ 139 & 154 \end{bmatrix} \end{align}$$$\boldsymbol{A}^3 = \boldsymbol{A} \times \boldsymbol{A} \times \boldsymbol{A}= \begin{bmatrix} 0 & 0 & 1 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{bmatrix} $
Följande vägar av längden 3 finns i $G$
Föraren jobbar på det lilla delproblemet och skriver koden
Navigatörens jobb är att aktivt granska koden som skrivs
Navigatörens jobb är inte att berätta för föraren exakt vad hen ska skriva, men kan komma med förslag
Byt roller ofta! Minst en gång varje halvtimme.