Lösungen

1

a

A={{2,-1,6},{4,4,1},{-1,5,3}}; A//MatrixForm

B={{2,2,1},{3,5,1},{-1,1,1}}; B//MatrixForm

C1=A.A; C1//MatrixForm

M=2 (A+B)-3 Transpose[(Transpose[A]-Transpose[B])]+ 5(C1-2A);
M//MatrixForm

b

F=Inverse[A].Inverse[B];
F//MatrixForm

F//N//MatrixForm

2

a

A={{2,-1,6},{4,4,1},{-1,5,3}}; 
Det[A]

b

Det[7 A]

c

1/Det[A]^100

1/Det[A]^100//N

3

a

A={{2,3,4},{3,4,5},{4,5,6}};A//MatrixForm

B={{2,1,0},{3,2,1},{4,3,2}};B//MatrixForm

A.B//MatrixForm

B.A//MatrixForm

A.B+B.A//MatrixForm

4

A={{2,-1,6},{x,4,x^2},{-1,5,3}};A//MatrixForm

Det[A]

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

N[%]

5

A={{1,0,1,2,5},{2,2,2,3,5},{3,2,1,-1,5},{-1,-2,1,1,5}}; b={-1,2,3,3};
A//MatrixForm

Solve[A.{x1,x2,x3,x4,x5}==b,{x1,x2,x3,x4,x5}]

6

Remove["Global`*"]

<<Calculus`VectorAnalysis`

?Grad

f[x_,y_]:=x^2+2y^2+4x-2y+3

Grad[f[x,y], Cartesian[x, y, z]]

a

Grad[f[x,y], Cartesian[x, y, z]]/.{x->3,y->2}

Delete[%,3]

b

Grad[f[x,y], Cartesian[x, y, z]]/.{x->3,y->2}

v=Delete[%,3]

u={4,-1}-{3,2}

u=u/Norm[u]

%//N

v.u

%//N

b1, Variante

u={4,-1}

u=u/Norm[u]

%//N

v.u

%//N

c

Solve[Grad[f[x,y], Cartesian[x, y, z]]=={0,0,0},{x,y}]

7

a

U=Transpose[{{1,-1,1},{1,-1,4},{-1,2,1}}];U//MatrixForm

Det[U]

b

Eigensystem[U]

%//N

Eigenvalues[U]

%//N

Eigenvectors[U]

%//N

U.Eigenvectors[U][[1]] / Eigenvalues[U][[1]]

c

Eigenvalues[U]

U.Eigenvectors[U][[1]] Eigenvalues[U][[1]]+U.Eigenvectors[U][[2]] Eigenvalues[U][[2]]+U.Eigenvectors[U][[3]] Eigenvalues[U][[3]]

%//N