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Notes 7: The Inverse of a Matrix
We show how to solve certain systems of linear equations. The ingredients of this
approach are two. First is the idea of the inverse function to a function. The second idea
is Gauss elimination and the ideas around Gauss elimination such as rank. These ideas
will be used often.
Deﬁnition 1.
Let
F
:
S
→
T,
G
:
T
→
S
be two functions. Assume that
G
◦
F
=
Id
S
and
F
◦
G
=
Id
T
, that is, for all
s
∈
S,G
◦
F
(
s
) =
s
and for all
t
∈
T,F
◦
G
(
t
) =
t
. We then say that
F
is the inverse function
to
G
and
G
is the inverse function to
F
.
Why are we interested in inverse functions?
Answer: If we are given an equation
F
(
x
) =
a
to solve and if
G
is the the inverse to
F
we
can apply it to the equation to get
G
◦
F
(
x
) =
G
(
a
)
x
=
G
(
a
)
.
Example
1
.
Given
f
:
R
→
R
x
7→
mx
m
∈
R
,m
6
= 0
,
then the inverse to
f
is the function
g
(
x
) =
m

1
x
so the solution to the equation
mx
=
a
is
x
=
g
(
a
) =
m

1
a.
Example
2
.
The function
f
:
R
→
R
x
7→
x
2
does not have an inverse because
f
only takes on positive values. If we start with
x
=

1
then there is no way to deﬁne
g
(

1) while staying in the real numbers so that
f
◦
g
(

1) =

1
.
Let
R
≥
0
denote the set of nonnegative real numbers.
Example
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