Unit 16
Matrix Equations and Inverses
Definition 17.1.
A set of m linear equations in n variables:
a
11
x
1
+
a
12
x
2
+
· · ·
a
1
n
x
n
=
b
1
a
21
x
1
+
a
22
x
2
+
· · ·
a
2
n
x
n
=
b
2
.
.
.
.
.
.
.
.
.
a
m
1
x
1
+
a
m
2
x
2
+
· · ·
a
mn
x
n
=
b
m
may be reexpressed in its
matrix form
as
a
11
a
12
a
13
· · ·
a
1
n
a
21
a
22
a
23
· · ·
a
2
n
.
.
.
.
.
.
.
.
.
.
.
.
a
m
1
a
m
2
a
m
3
· · ·
a
mn
x
1
x
2
x
3
.
.
.
x
n
=
b
1
b
2
b
3
.
.
.
b
n
i.e. has the form
Ax
=
b
Where A is called the
coeﬃcient matrix
of the system. So solving the origi
nal system is equivalent to solving
Ax
=
b
for the vector
x
. To this end, we
introduce the
inverse
of a matrix.
Definition 17.2.
Let
A
be a square matrix. If there exists a matrix
B
with
the same dimensions as
A
such that
AB
=
BA
=
I
We say that
A
is
invertible
(or
nonsingular
) and that
B
is the
inverse
of
A
, written
B
=
A

1
. If
A
has no inverse it is said to be
noninvertible
(or
singular
).
1
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Example 1.
If
A
=
2
−
1
0
1
show that
A

1
=
B
=
1
/
2
1
/
2
0
1
Solution:
We must show that
AB
=
BA
=
I
:
AB
=
2
−
1
0
1
1
/
2
1
/
2
0
1
=
(2)(1
/
2) + (
−
1)(0)
(2)(1
/
2) + (
−
1)(1)
(0)(1
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 Spring '10
 lit
 Calculus, Linear Equations, Equations, Invertible matrix

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