This is a minor extension of Slutsky’s theorem as it applies to matrices that are asymptotically invertible (i.e., converging to an invertible matrix, without necessarily being invertible for all $n$).
Theorem: Suppose $\y_n \inD \y$, where $\y$ is a $d \times 1$ random vector, $\A_n \inP \A$, where $\A$ is a positive definite matrix, and that $\y_n = \A_n\x_n$. Then
\[\x_n \inD \A^{-1}\y\]Proof: Given as homework. Note that the result would be trivial if $\A_n$ were invertible, but we don’t know that it is — only that its limit, $\A$, is invertible.