4.5
Inverse of a Square Matrix
Multiplicative identity
•
we know that
1
⋅
a
=
a
⋅
1 =
a
(for all
real numbers
a
)
•
multiplying any number by 1 yields the
identical
number
•
“1” is called the
multiplicative
identity element
for the
real number system
Now note:
=
=
d
c
b
a
d
c
b
a
d
c
b
a
1
0
0
1
1
0
0
1
so the system of 2
×
2 matrices also has an identity
element, called the
identity matrix
, denoted by “I”
In fact, the system of 3
×
3 matrices has its own identity
matrix, also denoted by I.
When I is mentioned, its
dimension must be inferred from context.
Multiplicative inverse
•
also, for any
real
number
a
≠
0
•
there is an element
a
1
, such that
a
⋅
a
1
=
a
1
⋅
a
= 1
•
a
1
is called the
multiplicative inverse
of
a
Q
: does every square matrix A have a multiplicative
inverse?
That is, a matrix A
1
such that AA
1
= A
1
A = I
?
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 Spring '11
 Staff
 Real Numbers, Invertible matrix, Identity element

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