Do Matrix Inverses Commute When Flanking Another Matrix?

[Woolyabyss]

I know that matrix and its inverse are commutative i.e. E*E^(-1) = E^(-1)*E

but is a matrix and its inverse at either side of another matrix commutative?
E*A*E^(-1) = E^(-1)*A*E

Any help would be appreciated.

 

[mfb]

Not in general.
Otherwise every coordinate transformation would be equivalent to its inverse transformation.