Let $A$ denote a unital commutative ring. Given a system of ideals $(I_p)_{p\in P}$ indexed by a partially ordered set $P$ such that if $p \leq q$, then $I_p$ is contained in $I_q$, when is

$$GL_n \left(\lim_{\leftarrow} (A/{I_p}) \right) \cong \lim_{\leftarrow} \ GL_n(A/{I_p}) ?$$

Of course this is true when $A$ is $I$-adically complete.

What happens when $A$ is a topological ring and the $GL_n$ 's are endowed with "suitable" topologies making them toplogical groups?