Simplify the matrix product to the identity

[hellokitten]

Homework Statement

IF G, H and G+H are invertible matrices and have the same dimensions
Prove that G(G^-1 + H^-1)H(G+H)^-1 = I

3. Attempt
G(G^-1 +H^-1)(G+H)H^-1 = G(G^-1G +G^-1H + H^-1G + H^-1H)H^-1
= (GG^-1GH^-1 +GG^-1HH^-1 +GH^-1GH^-1 +GH^-1HH^-1) = GH^-1+I +GH^-1GH^-1 +GH^-1
=2GH^-1+ GH^-1GH^-1

I am not sure where to go from here.

 

[LCKurtz]

Homework Statement

IF G, H and G+H are invertible matrices and have the same dimensions
Prove that G(G^-1 + H^-1)H(G+H)^-1 = I

3. Attempt
G(G^-1 +H^-1)(G+H)H^-1 = G(G^-1G +G^-1H + H^-1G + H^-1H)H^-1
= (GG^-1GH^-1 +GG^-1HH^-1 +GH^-1GH^-1 +GH^-1HH^-1) = GH^-1+I +GH^-1GH^-1 +GH^-1
=2GH^-1+ GH^-1GH^-1

I am not sure where to go from here.

Try starting with multiplying out what I have highlighted in red, then multiply the blue into that and see what it looks like.

 

[ehild]

Homework Statement

IF G, H and G+H are invertible matrices and have the same dimensions
Prove that G(G^-1 + H^-1)H(G+H)^-1 = I

3. Attempt
G(G^-1 +H^-1)(G+H)H^-1

Where did that starting expression come from?
Just expand the multiplications from left to right. What is G(G^-1+H^-1)?
Multiply the result with H from the right . What do you get?

LCKurtz was faster... :D