Matrix Simplification: A = HG(FHG)^{-1}FG | Step-by-Step Solution

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The discussion focuses on simplifying the matrix equation A = HG(FHG)^{-1}FG. The initial attempt correctly identifies that (FHG)^{-1} can be expressed as F^{-1}H^{-1}G^{-1}. The simplification process leads to the identity matrices I, which are ultimately eliminated, resulting in A = G. A participant initially questions the disappearance of the identity matrices but later realizes that multiplying by the identity matrix does not change the value of the matrix. The final answer confirmed in the discussion is A = G.
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Homework Statement



Simplify

A = HG(FHG)^{-1}FG

Homework Equations



None

The Attempt at a Solution



Well (FHG)^-1 is really just F^-1 H^-1 G^-1
Therefore A = HGG^{-1}H^{-1}F^{-1}FG
GG^-1 = I
FF-1 = I

Therefore A = HIH^{-1}(IG)

The book now simplifies to HH^{-1}G

I understand all the steps and the ending step, but I don't get how they got rid of the two I's

I X I = I (Identity Matrix), so where did it disappear to?

The final answer is: A = G

Thanks
 
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Urgh, silly me, solved.

Identity matrix multiplied by another matrix just returns that matrix I forgot.

Silly problem. Sorry.
 
JFonseka said:
Well (FHG)^-1 is really just F^-1 H^-1 G^-1

Not really. (FHG)^-1 = (G^-1)*(H^-1)*(F^-1).
 

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