Properties of The Inverse Matrix:
The inverse matrix of an n  n nonsingular matrix A has the following
important properties:
1.
2.
A 
A 
1
1
t
1
 A.

 A1

t
3. If A is symmetric, So is its inverse.
4.
 AB 1  B 1 A1
5. If C is an invertible matrix, then
甲、
AC  BC  A  B.
CA  CB  A  B .
乙、
6. As
 A  I 1 exists, then



I  A  A 2    A n1  A n  I  A  I    A  I  A n  I
1
1
[proof of 2]
A  A


  

1 t
t
 AA1
t
 It  I
similarly,
t
t
At A1  A1 A  I t  I
[proof of 3:]
By property 2,
A   A 
t 1
1 t
1
 A1 .
.
.
[proof of 4:]
B 1 A1  AB  B 1 A1 AB  B 1IB  I .
Similarly,
 ABB 1 A1  ABB 1 A1  AIA1  I
.
[proof of 5:]
Multiplied by the inverse of C, then
ACC 1  AI  A  BCC 1  BI  B .
Similarly,
C 1CA  IA  A  C 1CB  IB  B .
[proof of 6:]
I  A  A
2


   An1  A  I   A  A 2    An  I  A  A 2    An1
 An  I .
Multiplied by
 A  I  1
on both sides, we have


  A  I  A  I 
1  A  A 2    A n 1  A n  I  A  I 
I  A  A2    An1
can be obtained by using similar procedure.
Example:
Prove that
I  AB1  I  AI  BA 1 B .
[proof:]
2
1
n
1
.

I  AI  BA BI  AB  I  AB  AI  BA B  AI  BA  BAB
 I  AB  AI  BA   I  BA  BA B
 I  AB  AI  BA  I  BA B
1
1
1
1
1
1
 I  AB  AIB  I  AB  AB  I
Similar procedure can be used to obtain
I


 AB  I  AI  BA  B  I
1
3