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2. Inverse Matrizen

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by

Martin Weckerle

on 17 November 2018

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Transcript of 2. Inverse Matrizen

_
Inverse Matrizen
1 2
3 4
(
)
.
bla bo
bli bu
(
)
1. Berechnung

Inverser Matrizen
1 2
3 4
a b
c d
1 0
0 1
I.
1a
+
2c
=
1
II.
1b
+
2d
=
0
(
)
(
)
(
)
III.
3a
+
4c
=
0
IV.
3b
+
4d
=
1
.
=
1a + 2c = 1
3a + 4c = 0
1b + 2d = 0
3b + 4d = 1
a = -2 c = 1,5
-2 1
1,5 -0,5
(
)
a b
c d
(
)
=
Inverse Matrix:
2. Gauß - Jordan - Verfahren
(
)
A

A

=
E
.
-1
Übung: Berechne die Inverse
1 0
0 1
)
(
nur quadratische Matrizen!
2 1
1,5 0,5
-
-
Übung: Berechnung einer Inversen
(
)
1 2
3 4
(
)
(
)
2 1
1,5 0,5
-
-
(
)
2 1
1,5 0,5
-
-
1 2
3 4
(
)
-1
-1
=
=
1 2
a
c
1
1
0
a
b
c
d
1 2
1 2
3 4
a
b
c
d
0
0
3 4
a
b
c
d
0
1
b = 1 d = -0,5
1 0
0 1
3 4
b
d
(A ) = A
-1
-1
3 6
2 4
=
1 0
0 1
(
)
bla bo
bli bu
Inverse Matrix
1 0
0 1
Einheitsmatrix
1 0
0 1
(
)
(
)
(
)
1 0
0 1
Ziel:
1 2
3 4
1 2
0
3
1 0
(II)' = (II) + (-3) (I)
2
3 1
-
-
0
3
2
0
0 2
3 1
-
-
(I)' = (I) + (II)
1
2 1
-
2
1
2
-
0
(II)'' = (- ) (II)'
1
2
_
0
1
_
2 1
-
0
3
1
2
2
-
Inverse:
-2 1
1,5 -0,5
(
)
1 0
0 1
(
)
2
-
(
)
2 4
(
)
1,5
0
0,25
0,75
1
(II)' = 0,75 (I) + (II)
2 1
-
1 0
1
1,5
0
0
0,25
0,75
1
0
(I)' = (-4)(II)' + (I)
-
-
(II)'' = 4 (II)'
(
)

0 1
1
0
1
0,25
-
-
2 4
2 0
-
3 4
0
0,25
1
2
-
1
3 4
(I)'' = (I)'
1
2
_
1 2
0
)
(
(II)' = (- ) (I) + (II)
2
3
_
2
0
1 0
3 6
0
2
3
_
-
1
0
Nicht zu jeder Matrix existiert eine Inverse!
(II)
1,5 =
(I)
.
3 6
2 4
0
0
linear abhängig
A
A

=
E
-1
.
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