The limit of finite dimensional linear system of equations

Question

Consider the multiplication operator $$K_n:L^2(0,1)^n \rightarrow L^2(0,1)^n, \\ V(x)=(v_1(x),...,v_n(x)) \mapsto A_nV(x) ,$$ where $n$ is a positive integer and $A_n$ is a constant matrix of size $n$. Let us assume that $A_n$ is invertible for any $n$ with $\det(A_n)\neq0, \forall n\geq1$ and therefore our operator is invertible.

My question is the following: Do we have the invertibility of $K_n$ if $n\rightarrow\infty$?. How to prove such a result?. Actually, I conjecture that the answer is yes but I can not see a way to do it.

Any ideas?. Thank you in advance.

Answer 1

Take $n=1$, $K_n = 1/n$. Then $K_n\to0$, which is not invertible.