Lösungen

1

Remove["Global`*"]

a

P={{1,1},{0,β}}; P//MatrixForm

P.P//MatrixForm

P.P.P//ExpandAll//MatrixForm

P.P.P.P//ExpandAll//MatrixForm

b

Q=Inverse[P]; Q//MatrixForm

Q.Q//MatrixForm

Q.Q.Q//ExpandAll//MatrixForm

Q.Q.Q.Q//ExpandAll//MatrixForm

c

H={{u,v},{0,w}}; H//MatrixForm

hilf=H.Transpose[H]; hilf//MatrixForm

Solve[hilf=={{2,4},{4,16}},{u,v,w}]

H={{1,1},{0,4}}; H//MatrixForm

hilf=H.Transpose[H]; hilf//MatrixForm

2

a

i

A={{5,-1,3},{3,5,1},{-1,3,α}};A//MatrixForm

B=Transpose[{{1,1,1}}];B//MatrixForm

A.B//MatrixForm

Solve[A.B==Transpose[{{7,9,5}}],{α}]

ii

(A.B/.α->5)//MatrixForm

iii

Det[A]

Solve[Det[A]==0,{α}]

b

i

B={{5,-1,3},{3,5,1},{-1,3,-1}};B//MatrixForm

k[t_]:=Transpose[{{-4t,t,7t}}];
m[t_]:=Transpose[{{x,y,z}}];

hN[t_]:=B.k[t]; hN[t]//MatrixForm

ii

RowReduce[B]//MatrixForm

3

I^(1/6)

I^(1/6)//N

a

Remove["Global`*"]

solv3=Solve[z^6==I,{z}]//Flatten

%//N

tabZ = Table[zA[u]=z/.solv3[[u]],{u,1,6}]

b

tabPoints=Prepend[ Table[Point[{Re[zA[u]],Im[zA[u]]}],{u,1,6}],PointSize[0.03]]

Show[Graphics[tabPoints],AspectRatio->Automatic];

tabPoints=Table[{Re[zA[u]],Im[zA[u]]},{u,1,6}]

ListPlot[tabPoints,PlotStyle->{PointSize[0.03]},AspectRatio->Automatic];

c

Plus@@tabPoints

Apply[Plus,tabPoints]

Plus[tabPoints]

d

z1=E^(I Pi/2/6)//N

z1^5

z1^5 = zA[[4]]

1/z1

Nicht in der Liste der z[[u]], jedoch die Koordinaten sind in der Liste

Conjugate[z1]

z1+1/z1

Liegt auf der reellen Achse

4

Remove["Global`*"]

a

i

z[a_]:= a+I; w[a_]:=1/z[a]; Solve[z[a]+w[a]==0,{a}]

Solve[(a+I)^2+1==0,{a}]

a = 0 ist die einzige reelle Lösung. z ist dann gleich i.

ii

Solve[(a+I)^2+r==0,{a}]

b

Remove[z]

i

p1={4,1,-1};p2={3,-4,8};
q1={5,15,2};q2={8,-11,z};
{p1,p2,q1,q2}//Transpose//MatrixForm

Det[{p2-p1,q1-p1,q2-p1}]

slv=Solve[Det[{p2-p1,q1-p1,q2-p1}]==0,{z}]//Flatten;
z=z/.slv

%//N

{p1,p2,q1,q2}//Transpose//MatrixForm

ii

g1[t1_]:=p1+t1 (p2-p1);
g2[t2_]:=q1+t2 (q2-q1);
slv1=Solve[g1[t1]==g2[t2],{t1,t2}]//Flatten

{t1,t2}={t1,t2}/.slv1

g1[t1]

s1=g2[t2]

g2[t2]//N

iii

cr=Cross[p1-s1,q1-s1]

Norm[cr]/2

%//N