Simplify the matrix product to the identity

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SUMMARY

The discussion centers on proving the equation G(G^-1 + H^-1)H(G+H)^-1 = I, where G and H are invertible matrices of the same dimensions. Participants provide a detailed attempt at expanding the left-hand side of the equation, leading to the expression 2GH^-1 + GH^-1GH^-1. The conversation highlights the importance of systematic multiplication and expansion of matrix products to simplify complex expressions. Suggestions for further steps include focusing on the multiplication of highlighted terms to clarify the proof.

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hellokitten
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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.
 
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hellokitten said:

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.
 
Last edited by a moderator:
hellokitten said:

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
 

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