Lösungen

1

(A * X * (A^(-1)) ) (A * X * 
              (A^(-1)) )  +   (A^T)  * X * (A^(-1)) = 
      M -  (A^(-1)) + (M^T) + A * ((X )^2)* (A^(-1))

A * X * X * (A^(-1))  +   ((A^T)  * X * (A^(-1))) = 
      M -  (A^(-1)) + (M^T) + A * ((X )^2)* (A^(-1))

(A^T)  * X * (A^(-1)) = M -  (A^(-1)) + (M^T)

(A^T)  * X  = M*A -  (A^(-1))*A + (M^T)*A

X  = ((((A^T)))^(-1))*M*
          A -  ((((A^T)))^(-1)) + ((((A^T)))^(-1))*(M^T)*A

X  = ((((A^T)))^(-1))*(M*A + (M^T)*A -  E)

X  = ((((A^T)))^(-1))*((M + (M^T))*A -  E)

X  = ((((A^T)))^(-1))*( (A^(-1))*(A^T)  *A -  E)

2

Remove["Global`*"]

v1[
          t_] := {1, 
            2, -1} + λ {2, 1, 1};
v2[t_] := {-1, 1, 1} + μ {4, 5, -2};

V = 
      Det[{{-1, 1, 1} - {1, 2, -1}, {2, 1, 1}, {4, 
            5, -2}}]

A = Norm[Cross[{2, 1, 1}, {4, 5, -2}]]

d = (V/A)

d = (V/A) // N

3

Remove["Global`*"]

OA = {-1, 1, 1};  A = OA;

Φ[λ_, μ_] := 
        OA + λ {2, 1, 1} + μ {4, 
              5, -2};
Q = {10, -10, 12};

a

V = Det[{{2, 1, 1}, {4, 5, -2}, Q - {-1, 1, 1}}]

V/6

N[%]

b

d = V/Norm[Cross[{2, 1, 1}, {4, 5, -2}]]

-d // N

c

n = Cross[{2, 1, 1}, {4, 5, -2}]

g[t_] := Q + t n;
solv = 
      Solve[Φ[λ, μ] == g[t], {λ, μ, t}] // 
        Flatten

L = g[t] /. solv

N[%]

d

Q1 = Q + 2 (L - Q)

N[%]

e

ArcCos[(Q - A) . (Q1 - A)/(Norm[Q - A]  Norm[
              Q1 - A])]

N[%]

%/Degree

4

Remove["Global`*"]

OA = {-1, 1, 1};  A = OA;

Φ1[x_, y_, 
          z_] := (3  x + 5  y - 5  z - 12 == 
            0);
Φ2[x_, y_, 
          z_] := (-4 x + 5  y + 3  z + 12 == 
            0);
OQ = {1, 2, 10};

solv4 = 
      Solve[{Φ1[x, y, z], Φ2[x, y, z]}, {x, y, z}] // 
        Flatten

p[u_] := ({x, y, z} /. solv4) /. z -> u;   p[u]

n = 35 (p[1] - p[0])

n1 = {3, 5, -5};  n2 = {-4, 5, 3};
nn = Cross[n1, n2]

solv4a = 
      Solve[n . (OQ - p[u]) == 0, {u}] // Flatten

p[u] /. solv4a

N[%]

5

Remove["Global`*"]

a

Gl1 = (x + y + z == 1);
Gl2 = (2  x + 3  y + 4  z == 
            1);
Gl3 = (x + 2  y + 3  z == 1);
Solve[{Gl1, Gl2, Gl3}, {x, y, z}]

Keine Lösung

b

Gl1 = (x + y + z == 1);
Gl2a = (2  x + 3  y + 4  z == 
            2);
Gl3 = (x + 2  y + 3  z == 1);
Solve[{Gl1, Gl2a, Gl3}, {x, y, z}]

Lösung mit Dim = 1

6

Remove["Global`*"]

A = {{1, 2, 1}, {1, x, 2}, {-1, 
            2, -4}};

    B = {{1, 0, 3}, {2, 3, 1}, {1, 0, 1}};

a

Det[A]

Solve[Det[A] == 0, {x}]

N[%]

Det[B]

b

A . B // MatrixForm

c

(3 Inverse[A] . B /. x -> 1/3) // MatrixForm

d

(3 B . Inverse[A] /. x -> 1/3) // MatrixForm

e

(B . A . B . Inverse[A] /. x -> 1) // MatrixForm

f

(B . Transpose[B] . A . Transpose[A] /. x -> 1) // 
      MatrixForm

g

(Inverse[Transpose[Inverse[B] . Inverse[A]]] /. x -> 1) // 
      MatrixForm

7

Remove["Global`*"]

M = {{2, -2}, {-3, 1}};  
    OP0 = {5, 8};
  M // MatrixForm

MD[φ_] := {{Cos[φ], -Sin[φ]}, {Sin[φ], Cos[φ]}};
 MD[φ] // MatrixForm

a

MGesamt = Inverse[M] . MD[-12 Degree] . M;  
    MGesamt // MatrixForm

MGesamt // N // MatrixForm

b

OP1 = M . OP0

OP2 = MD[-12 Degree] . OP1

N[%]

OP3 = Inverse[M] . OP2

N[%]

MGesamt . OP0  // N

8

Remove["Global`*"]

p1 = {1, 1, 0};  p2 = {1, 0, 1};  
    p3 = {0, 1, 2};  s = (p1 + p2 + p3)/3

N[%]

a

n = Cross[p2 - p1, p3 - p1]

v = n/Norm[n]

-N[%]

b

g[t_] := s  +  t n;   g[t]

solv8 = Solve[g[t][[3]] == 15, {t}] // Flatten

P4 = g[t] /. solv8

N[%]

c

M = {v, p2 - p1, p3 - p1} // Transpose;
M // MatrixForm

N[%] // MatrixForm

d

M . (p1 + 2 (p2 - p1) + 2 (p3 - p1))

N[%]

e

Norm[Cross[p2 - p1, p3 - p1]]/2

N[%]

f

Norm[Cross[2 p2 - p1, 3 p3 - p1]]/2

N[%]