UNIT16 - Unit 16 Matrix Equations and Inverses Denition...

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 re-expressed 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

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