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1
Properties: For
n
= 1, the definition reduces to the multiplicative
inverse (
ab
=
ba
= 1).
If
B
is an inverse of
A
, then
A
is an inverse of
B
, i.e.,
A
and
B
are inverses to each other.
Example:
⇒
Example: Some matrices have no inverse, like
O
∈
R
n
×
n
and
since
Proof
Suppose
B
and
C
are both inverses of
A
.
Then
Notation:
The inverse of
A
, if exists, is denoted by
A
−
1
.
Symbolically, the inverse may be used to solve matrix equations:
A
x
=
b
However, this method is computationally inefficient.
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Example:
⇒
⇒
Proof
(a)
A
−
1
and
A
are the inverse to each other.
(b) (
AB
)(
B
−
1
A
−
1
) =
A
(
BB
−
1
)
A
−
1
=
AI
n
A
−
1
=
AA
−
1
=
I
n
;
similarly, (
B
−
1
A
−
1
) (
AB
) =
I
n
.
(c)
A
T
(
A
−
1
)
T
= (
A
−
1
A
)
T
= (
I
n
)
T
=
I
n
; similarly, (
A
−
1
)
T
A
T
=
I
n
.
then
A
T
is also invertible, and (
A
T
)
−
1
= (
A
−
1
)
T
.
2
Definition.
An
n
×
n
matrix
E
is called an elementary matrix if
E
can
be obtained from
I
n
by a single elementary row operation.
.
1
0
2
0
1
0
0
0
1
,
1
0
0
0
4
0
0
0
1
,
0
1
0
1
0
0
0
0
1
:
Example
3
2
1
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
E
E
E
.
10
7
4
3
2
1
6
5
4
3
2
1
1
0
2
0
1
0
0
0
1
6
5
4
3
2
1
:
Example
3
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
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 Spring '09
 Fong

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