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Remove["Global`*"]

0. Materialbereitstellung

A2 = {{1,2},{1,-1}}; 
B2 = {{3,5},{4,7}}; 
C2 = {{4,1},{5,6}};
Map[MatrixForm,{A2,B2,C2}]

A3 = {{1,2,3},{1,-1,0},{-2,1,5}}; 
B3 = {{3,5,1},{4,7,9},{3,2,6}}; 
C3 = {{4,1,3},{5,6,5},{5,8,8}};
Map[MatrixForm,{A3,B3,C3}]

A4 = {{1,2,3,4},{1,-1,0,1},{-2,1,5,2},{-2,2,1,5}}; 
B4 = {{3,5,1,1},{4,7,9,5},{3,2,6,8},{5,6,5,1}}; 
Map[MatrixForm,{A4,B4}]

A24 = {{3,5,1,1},{4,7,9,5}}; 
B42 = {{1,2},{1,-1},{-2,1},{-2,2}};  
Map[MatrixForm,{A42,B24}]

X13 = {{x11,x12,x13}}; 
X31 = {{x11},{x21},{x31}}; 
X24 = {{x11,x12,x13,x14},{x21,x22,x23,x24}}; 
X42 = {{x11,x12},{x21,x22},{x31,x32},{x41,x42}};  
Map[MatrixForm,{X13,X31,X24,X42}]

b31 = {{50},{-100},{1000}}; 
b32 = {{50,203},{-100,105},{1000,-50}}; ;  
Map[MatrixForm,{b31,b32}]

Em2 = IdentityMatrix[2]; 
Em3 = IdentityMatrix[3]; 
Em4 = IdentityMatrix[4]; 
Map[MatrixForm,{Em2,Em3,Em4}]

NullMatrix[m_]:= Table[ Table[0,{k,1,m}],{k,1,m}];
NullMatrix[4]//MatrixForm

1. Matrixaddition, Multiplikation mit Skalar

a

(A2+B2) // MatrixForm

(B2+C2) // MatrixForm

(A2+B2)+C2 // MatrixForm

A2+(B2+C2) // MatrixForm

(A3+B3)+C3 // MatrixForm

A3+(B3+C3) // MatrixForm

b

(A2+B2)+C2 == A2+(B2+C2)

(A3+B3)+C3

(A3+B3)+C3 == A3+(B3+C3)

c

4 A3+(-5) B3+ 6 C3 // MatrixForm

4 A3 -5 B3 - 6 C3 // MatrixForm

d

A4 + Em4 // MatrixForm

A4 + Em4 + NullMatrix[4]   // MatrixForm

e

Die Koeffizentenmatrizen werden addiert.

2. Matrixmultiplikation

a

A3.X31   // MatrixForm

b

A3.X13   // MatrixForm

Funktioniert nicht!!!

c

A4.X42   // MatrixForm

d

A24.X42   // MatrixForm

e

B42.X24   // MatrixForm

f

A3.b31   // MatrixForm

g

A3.b32   // MatrixForm

h

(A2.B2).C2   // MatrixForm

A2.(B2.C2)   // MatrixForm

h

(A3.B3).C3   // MatrixForm

A3.(B3.C3)   // MatrixForm

i

A3.B3   // MatrixForm

B3.A3   // MatrixForm

A3.B3 == B3.A3

k

A4.B4   // MatrixForm

B4.A4   // MatrixForm

A4.B4 == B4.A4

l

A3.Em3   // MatrixForm

Em3.A3   // MatrixForm

A4.Em4   // MatrixForm

Em4.A4   // MatrixForm

A4.Em4 == A4

Em4.A4 == A4

m

Det[A4]

Inverse[A4]   // MatrixForm

Inverse[A4].A4   // MatrixForm

A4.Inverse[A4]   // MatrixForm