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Inverse of a matrix to the nth power Is the nth Power of the Inverse of the Matrix: Proof by Sankofakanian
n
n -1
-1 n
and ( A ) = (A ) for n = 0,1,2,... If A is invertible, then A is invertible and proof: -1 1
1
1
-1
•
Let P(1): (AA ) = I = A A . P(1) is true.
•
Let k be any natural number. If P(k): (AA ) = A A
-1 k
-1 k+1
P(k+1): (AA ) -1 k+1
k+1
k
(AA )
-k-1
=A A -k
k
-k
is true then we must prove that
is also true. That would mean that
-1
= A A (AA ). -1
To ease our work, lets put a subscript i from {1,..,n} on each A and A depending on the -1
-1 k
-1 k+1
position of AA in the product (AA ) . Following this we can write (AA ) -1 k
-1
-1
as
-1
(AA ) = (A1A1 )(A2A2 )...(Ak Ak ) and -1 k+1
-1
(AA )
-1
-1
-1
= (A1A1 )(A2A2 )...(Ak Ak )(AA ) -1
-1
= A1(A1 A2A2 ... A k-1
-1
-1 k-1
-1
-1
Ak Ak )(AA )
-1
= (A1I Ak )(AA ) -1 k+1
-1
(AA )
k-2
-1
-1
= (A1(A2 A1 )I A k )(AA )
Now, Since A 1 = A2 = A, we have -1 k+1
2
(AA )
-1 k-2 1
= (A A
-1 k-2
Moreover, because A 1 I -1 k+1
(AA )
2 k-2
= (A I
-1
I
A
-1
Ak )(AA )
k-2
-1
= I A1 we have
-1 1
-1
-1
2
k-2
k-2
IA
Ak )(AA ) = (A I
-2
-1
-1
-1 1
A )(AA ) since A k = A
-1
=A
Now 2
k-2
(A I
-2
-1
2
k-2
A )(AA ) = A I 2
2
k-2
-2
-1
3
2
-2
2
-2
-2
-2
k-2
since A I = I A
AA A = A A I
k-2
A
A )(AA ) = A I
-1
-2
k-2
= A I (A I
-2
A I=A I
k-2
-1
A A since I
A= AI
k-2
-3
Now we will pursue the same process of rewriting by induction. -1 j
e j-e
-e
To do so Let Q(j): (AA ) = A I A be true for any any 0 =< e =< j . We want to prove http://www.scribd.com/full/ http://www.scribd.com/full/31221377?access_key 31221377?access_key=key-a60g164r0xzgt =key-a60g164r0xzgtihybqm ihybqm
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that e j+1-e
Q(j+1): A I -1 k+1
(AA )
e
-e
A is true for any 0 =< e =< j+1 and and k+1-e
=A I
-e
A is true for any 0 =< e =< k+1.
Hence for e = k+1 we have -1 k+1
(AA )
k+1
= A
0
I A
-k-1
k+1
-k-1
=A A
which proves that P(k+1) is true. Therefore by mathematical induction, P(n) is true for f or any natural number n and -1
n
n
( AA AA ) = A A n
-n
-n
-n
n
n
Now since A A = I = A A , A is invertible and by Laws of Exponent -n
Algebra, Ninth Edition 1 H. Anton, C. Rorres . Elementary Linear Algebra, 2 H. Anton, C. Rorres . Elementary Linear Algebra, Algebra, Ninth Edition http://www.scribd.com/full/ http://www.scribd.com/full/31221377?access_key 31221377?access_key=key-a60g164r0xzgt =key-a60g164r0xzgtihybqm ihybqm
