In 1977, Joan Plastiras gave a striking example of two non $\*$-isomorphic C$^\*$-algebras $\mathcal A$ and $\mathcal B$ such that $$\mathcal A \otimes M\_2(\mathbb C) \simeq \mathcal B\otimes M\_2(\mathbb C)$$ (<http://www.ams.org/journals/proc/1977-066-02/S0002-9939-1977-0461158-9/S0002-9939-1977-0461158-9.pdf>).


My question is this: can you find two non $\*$-isomorphic C$^\*$-algebras $\mathcal A,\mathcal B$ such that $$\mathcal A\otimes M\_n(\mathbb C) \simeq \mathcal B\otimes M\_n(\mathbb C), \ n=2,3?$$ 
Note that Plastiras' example does not satisfy this condition. Furthermore, can you find non $\*$-isomorphic $\mathcal A, \mathcal B$ such that
$$\mathcal A\otimes M\_n(\mathbb C) \simeq \mathcal B\otimes M\_n(\mathbb C), \ \forall n\geq 2$$
or does this condition imply that $\mathcal A\simeq \mathcal B$?



C$^*$-algebras isomorphic after tensoring with $M_n(\mathbb C)$
Such examples do exist. In the paper "Stability of C\*-algebras is not a stable property, Doc. Math. J. DMV , 2, (1997), 375-386.", Rordam gives examples of simple C\*-algebras $A$ such that $M\_2(A)$ is a stable C\*-algebra but $A$ is not. Now take the pair $A$ and $B=M\_2(A)$. These C\*-algebras are not isomorphic, since $A$ is not stable, but $B$ is. On the other hand, $M\_n(A)$ is isomorphic to $M\_n(B)$ which is isomorphic to $A\otimes \mathcal K$ for all $n\geq 2$. (Once $M\_n(A)$ is stable for some $n$, it remains stable for all larger $n$. This is proven in the same paper.)


Part of the hard work in Rordam's paper is to find an example as above with $A$ simple. At the heart of Rordam's construction is the following non-simple example: There exists a countable generated Hilbert C\*-module $H$ over $C(X)$, with $X=\prod\_{i=1}^\infty S^2$ (an infinite product of 2-spheres) such that $H$ is not isomorphic to $\ell^2(C(X))$
but $H\oplus H$ is. Then one can choose $A=K(H)$ and $B=K(H\oplus H)$. $A$ is not stable since this is equivalent to $H\cong \ell^2(C(X))$ but $M\_2(A)\cong B$ is stable.


This is only a comment:
It seems to me that it should be possible to find (hermitian) complex vector bundles $\eta\_1$ and $\eta\_2$ over some space $X$ such that $\eta\_1^{n\oplus}\cong \eta\_2^{n\oplus}$ for all $n\geq 2$ but $\eta\_1\ncong \eta\_2$. Then $A=\mathrm{End}(\eta\_1)$ and $B=\mathrm{End}(\eta\_2)$ are candidate examples. From $\eta\_1^{n\oplus}\cong \eta\_2^{n\oplus}$ for all $n\geq 2$ we can already deduce that $M\_n(A)\cong M\_n(B)$ for all $n\geq 2$. If we choose them with some care (they can't be line bundles) then we should also have that $A\ncong B$.


