Assume that $A$ is a unital $C^{*}$ algebra. Is there a subvector space $Y\subset A$ of finite codimension which does not contain any invertible element?

Let $n(A)$ be the infimum of such codimensions. For example $n(A)=1$ if $A$ is commutative. Or $n(M_{n}(\mathbb{C}))=n$.

For a commutative $A$, is it true to say $n(A\otimes M_{n}(\mathbb{C}))=n\times 1=n$? More generaly, can one express $n(A\otimes B)$ in term of $n(A)$ and $n(B)$?

Note: This question is somehow a reverse question to the following famous problem:

What is the maximim possible dimension for those subvector space of $M_{n}(\mathbb{R})$ which consist only invertible matrices(except zero matrix). For what valuse of $n$ the sharp dimension would occure?