Question about invertible matrices

  • Thread starter icesalmon
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  • #1
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Homework Statement


Show that if A, B and A + B are all invertible and the same size
then A(A-1 + B-1)B(A + B)-1 = I
And what does the result say about A-1 + B-1


The Attempt at a Solution


I start off by trying to reduce the LHS as much as I can so I multiply both sides on the right by ( A + B )
to get A(A-1 + B-1)B = (A+B)
(A-1 + B-1)B = A-1(A+B)
A-1 + B-1 = A-1(A+B)B-1
A-1 + B-1 = (A-1A + A-1B)B-1
A-1 + B-1 = ( IB-1 + A-1I )
A-1 + B-1 = A-1 + B-1
:confused:
I've gotten the same thing on both sides, and yet again I don't even know what i've done.
thanks PF
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
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Homework Statement


Show that if A, B and A + B are all invertible and the same size
then A(A-1 + B-1)B(A + B)-1 = I
A direct calculation on the left gives (I+ AB-1)B(A+B)-1= (B+ A)(A+ B)-1. And, of course, matrix addition is commutative.

And what does the result say about A-1 + B-1
Well, it says it is the inverse of ....


The Attempt at a Solution


I start off by trying to reduce the LHS as much as I can so I multiply both sides on the right by ( A + B )
to get A(A-1 + B-1)B = (A+B)
(A-1 + B-1)B = A-1(A+B)
A-1 + B-1 = A-1(A+B)B-1
A-1 + B-1 = (A-1A + A-1B)B-1
A-1 + B-1 = ( IB-1 + A-1I )
A-1 + B-1 = A-1 + B-1
:confused:
I've gotten the same thing on both sides, and yet again I don't even know what i've done.
thanks PF
 
  • #3
35,237
7,057

Homework Statement


Show that if A, B and A + B are all invertible and the same size
then A(A-1 + B-1)B(A + B)-1 = I
And what does the result say about A-1 + B-1


The Attempt at a Solution


I start off by trying to reduce the LHS as much as I can so I multiply both sides on the right by ( A + B )
If you do this, you are tacitly assuming that the equation is a true statement. Instead, work with the expression on the left side to show that it is equal to I.
to get A(A-1 + B-1)B = (A+B)
(A-1 + B-1)B = A-1(A+B)
A-1 + B-1 = A-1(A+B)B-1
A-1 + B-1 = (A-1A + A-1B)B-1
A-1 + B-1 = ( IB-1 + A-1I )
A-1 + B-1 = A-1 + B-1
:confused:
I've gotten the same thing on both sides, and yet again I don't even know what i've done.
thanks PF
 
  • #4
270
13
well, it says it is the inverse of ....
(a-1 + b-1)-1(a-1 + b-1)1 = (a-1 + b-1)0 = I
so I would say, (a-1 + b-1)-1
what do you think? Thanks you for your help.
 
  • #5
270
13
If you do this, you are tacitly assuming that the equation is a true statement. Instead, work with the expression on the left side to show that it is equal to I.
I know i've done this in other threads, so it must seem like i'm not listening, I just get so careless. Thanks for the help
 

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