Signals and Systems with MATLAB
Computing and Simulink
Modeling, Fourth Edition
D
−
21
Copyright
©
Orchard Publications
The Adjoint of a Matrix
D.8 The Adjoint of a Matrix
Let us assume that
is an
n
square matrix and
is the cofactor of
. Then
the adjoint of
,
denoted as
,
is defined as the
n
square matrix below.
(D.41)
We observe that the cofactors of the elements of the ith row (column) of
are the elements of
the ith column (row) of
.
Example D.12
Compute
if Matrix
is defined as
(D.42)
Solution:
D.9
Singular and Non
−
Singular Matrices
An
square matrix
is called
singular
if
;
if
,
is called
non
−
singular
.
A
α
ij
a
ij
A
adjA
adjA
α
11
α
21
α
31
… α
n
1
α
12
α
22
α
32
… α
n
2
α
13
α
23
α
33
… α
n
3
…
…
… …
…
α
1
n
α
2
n
α
3
n
… α
nn
=
A
adjA
adjA
A
A
1
2
3
1
3
4
1
4
3
=
adjA
3
4
4
3
2
3
4
3
–
2
3
3
4
1
4
1
3
–
1
3
1
3
2
3
3
4
–
1
3
1
4
1
2
1
4
–
1
2
1
3
7
–
6
1
–
1
0
1
–
1
2
–
1
=
=
n
A
detA
0
=
detA
0
≠
A

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
Appendix D
Matrices and Determinants
D
−
22
Signals and Systems with MATLAB
Computing and Simulink
Modeling, Fourth Edition
Copyright
©
Orchard Publications
Example D.13
Matrix
is defined as
(D.43)
Determine whether this matrix is singular or non
−
singular.
Solution:
Therefore, matrix
is singular.
D.10
The Inverse of a Matrix
If
and
are
square matrices such that
, where
is the identity matrix,
is
called the
inverse
of
, denoted as
, and likewise,
is called the inverse of
, that is,
If a matrix
is non-singular, we can compute its inverse
from the relation
(D.44)
Example D.14
Matrix
is defined as
(D.45)
Compute its inverse, that is, find
A
A
1
2
3
2
3
4
3
5
7
=
detA
1
2
3
2
3
4
3
5
7
1 2
2 3
3 5
21
24
30
27
–
20
–
28
–
+
+
0
=
=
=
A
A
B
n
AB
BA
I
=
=
I
B
A
B
A
1
–
=
A
B
A
B
1
–
=
A
A
1
–
A
1
–
1
detA
------------adjA
=
A
A
1
2
3
1
3
4
1
4
3
=
A
1
–