| Nonsingular
Matrices
A
matrix A is said to be nonsingular (or invertible) if there
exists a square matrix B such that :
A
B = I
B
A = I
The
matrix B is then called an inverse of A and is denoted
by A-1
|
|
| I.
Some Properties of Nonsingular Matrices |
(1) If A
and B are nonsingular matrices of the same order, then
AB is
nonsingular and ( AB )-1
= B-1A-1 |
(2) If A
is nonsingular, then Ak is nonsingular for any positive
integer k
and ( Ak )-1=
( A-1)k. |
| (3) If A-1is
a nonsingular matrix, then ( A-1
)-1=
A . |
(4) If A
is nonsingular and k is a nonzero scalar, then kA is nonsingular
and ( kA )-1=
(1/k) A-1.
|
|
| Theorem
I : |
|
Let
A be a n x n nonsingular
matrix, then
(i)
the homogeneous system Ax = 0 has only the trivial
solution
(ii)
the system Ax
= b has
a unique solution for any b in Rn
|
|
| |
| II.
Elementary Matrix |
A n
x n matrix is called an elementary matrix if by performing
an appropriate
single elementary row operation, it becomes an identity matrix. |
|
Example 1 :
|
E1
= 
is
an elementary matrix because it becomes an identity matrix
by simply interchanging Row 2 and Row 3.
|
|
Example
2 :
|
E2
= 
is
an elementary matrix because by performing Row 3 + 2(
Row 1 ) and putting the result back to Row 3, E2
becomes an identity matrix.
|
A useful
operation on elementary matrix is that :
If E
is a n x n elementary matrix and A is a n
x m matrix, E A is the matrix
that results when this same elementary row operation is performed.
|
Example
3 :
|
If
E1 is as defined in Example 1 and
A
=
E1 A
=
|
|
Example
4 :
|
If
E2 is as defined in Example 2 and
E2 A
=
is
the same as R3 - 2R1
 R3
performing on A
|
Note that
every elementary matrix is nonsingular and the inverse of an
elementary matrix is also an elementary matrix.
|
Example
5 :
|
An
involution matrix is that
A
=
A
-1
or
A2 =
I
or
A2n =
I
for
all positive integer n.
E1
in Example 1 is obviously an involution matrix because
because E1 ( E1 A
) = ( E1 E1 ) A =
A
Implies E1 = ( E1 )
-1
Can
you give another involution matrix ?
Answer
:
|
|
|
| III.
Finding Inverse |
| Theorem
II : |
|
Let
A be a square
matrix of order n, then the following
statements are equivalent :
(i)
A is nonsingular
(ii)
Ax = 0 has only the trivial solution
(iii)
A can be reduced to I
by
elementary row operations
|
|
|
The
theorem gives that if A can be reduced to the identity
matrix by
using elementary row operations P1 , P2
, ....... , Pk that is
A  A1  A2
. . . . . . . . Ak-1 Ak
= I
|
|
and
if Ej
is the elementary matrix by applying Pj
to I
, then
E1A
=
A1 , E2E1A
= E2A1
= A2 ,
. . . . ., Ek...E2E1A
= Ak
=
I
|
|
As
a result, A-1
= Ek
. . . .E2E1
I
Thus,
to find the inverse of A, we can find a sequence
of elementary
row operations that reduces A to I,
and then perform this same
sequence of elementary operations on I to obtain
A-1
.
Symbolically,
we have
[
A I I
]
[ A1 I E1I
]
[ A2 I E2E1I
]
. . .
[ I
I Ek...E2E1I
]
= [ I
I
A-1
]
|
|
|
Remark
:
|
If
we apply the above method to a singular matrix A
, it is impossible to reduce A to the identity
matrix because at some point in the reduction of [ A
| I
] , a zero
row will be introduced. Hence we can conclude that the
given matrix A is singular.
|
We conclude
the nonsingular matrix by stating an important theorem.
| Theorem
III : |
|
Let
A be a any n
x n matrix. The following statements are
equivalent :
(i)
A is nonsingular
(ii)
The homogeneous system Ax = 0 has only the
trivial
solution
(iii)
A can be reduced to I
by
a sequence of elementary row
operations
(iv)
The non-homogeneous system Ax = b is consistent
for every
vector b in
Rn
|
|
|
Exercise
1
|
Give
the elementary matrix E1 that results
Row 3 - Row 1
Row 3 of an order 3 matrix.
Answer
:
|
|
Exercise
2
|
Give
the elementary matrix E2 that results
Row 2 - 1/2 (Row 3)
Row 2 of an order 3 matrix.
Answer
:
|
|
Exercise
3
|
Give
the elementary matrix E3 that results
dividing Row 2 by 5 and Row 3 by 8 of an order 3 matrix.
Answer
:
|
|
Exercise
4
|
Give the result E3
E2 E1
Answer
:
|
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