Lecture 6
Andrei Antonenko
February 12, 2003
1 Multiplicative inverse
In this lecture we will continue with the properties of matrix operations.
(M4) Existence of the multiplicative inverse.
Matrix
B
is called the
inverse
for the
square matrix
A
if
BA
=
AB
=
I
. The existence of the inverse is a very diﬃcult
question, which we will solve later. Now we’ll simply give some examples, and then a
formula for an inverse of 2
×
2matrices.
Example 1.1.
The matrix
ˆ
1 0
0 0
!
doesn’t have an inverse, since if
ˆ
a b
c d
!
is an inverse,
than
ˆ
1 0
0 0
!ˆ
a b
c d
!
=
ˆ
1 0
0 1
!
, from what
ˆ
a b
0 0
!
=
ˆ
1 0
0 1
!
which is never the case.
Example 1.2.
The inverse for the matrix
ˆ
1 2
3 5
!
is
ˆ

5
2
3

1
!
since
ˆ
1 2
3 5
!ˆ

5
2
3

1
!
=
ˆ
1 0
0 1
!
Now we’re ready to give a formula for an inverse of 2
×
2matrix.
Proposition 1.3.
The inverse of
2
×
2
matrix
ˆ
a b
c d
!
exists if and only if
ad

bc
6
= 0
and
ˆ
a b
c d
!

1
=
1
ad

bc
ˆ
d

b

c
a
!
Proof.
If
ad

bc
6
= 0 then we can simply check that this matrix is the inverse:
ˆ
a b
c d
!
×
1
ad

bc
ˆ
d

b

c
a
!
=
1
ad

bc
ˆ
ad

bc

ab
+
ab
cd

cd

cb
+
da
!
=
ˆ
1 0
0 1
!
1
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View Full DocumentWe will not give a proof that if a matrix has an inverse, then
ad

bc
6
= 0. This fact can be
generalized to the case of larger matrices, and we’ll prove it later in more general form.
The proposition above is useful when you want to get an inverse of a 2
×
2matrix. Later
we’ll provide a method of ﬁnding the inverses of larger matrices, but for 2
×
2matrices this is
the easiest one.
2 Transpose of a matrix
Deﬁnition 2.1.
Matrix
B
is called
transpose
of a matrix
A
(notation:
B
=
A
>
) if
b
ij
=
a
ji
.
In other words, we should take rows of a matrix
A
and write them as columns of matrix
B
.
Then
B
=
A
>
. In general form, if
A
=
a
11
a
12
···
a
1
n
a
21
a
22
···
a
2
n
.
.
.
.
.
.
.
.
.
.
.
.
a
m
1
a
m
2
···
a
mn
then
A
>
=
a
11
a
21
···
a
m
1
a
12
a
22
···
a
m
2
.
.
.
.
.
.
.
.
.
.
.
.
a
1
n
a
2
n
···
a
mn
Let’s notice, that if
A
is an
m
×
n
matrix, then
A
>
is an
n
×
m
matrix.
Example 2.2.
Let
A
=
ˆ
1 2
3 4
!
. Than
A
>
=
ˆ
1 3
2 4
!
.
Example 2.3.
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 Spring '03
 Andant
 Linear Algebra, Invertible matrix, Elementary matrix

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