Question about invertible matrices

[icesalmon]

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

I've gotten the same thing on both sides, and yet again I don't even know what I've done.
thanks PF

 

[HallsofIvy]

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

I've gotten the same thing on both sides, and yet again I don't even know what I've done.
thanks PF

 

[Mark44]

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

I've gotten the same thing on both sides, and yet again I don't even know what I've done.
thanks PF

 

[icesalmon]

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.

 

[icesalmon]

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