Assume that $A$ is  a unital $C^{*}$ algebra. Is there a subvector space $Y\subset A$ 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 is there any relation between $n(A\otimes B)$ and $n(A)$ and $n(b)$?