I'm sorry, I don't understand what you mean by "dim(A)". A matrix is NOT a vector space and so does not have a dimension. It also does NOT follow that if A is invertible, then det(A)= 1. If A is invertible then dim(A) is not 0 but can be any non-zero number.
If A is invertible, then its rank is equal to the dimension of its domain space and so is the rank of , , etc.