Let $A$ and $B$ be two $C^{\ast}$-algebras such that their matrix algebras, $M_2(A)$ and $M_2(B)$, are $\ast$-isomorphic $C^\ast$-algebras.
Question 1: Are $A$ and $B$ isomorphic $C^\ast$-algebras?
In a related case, let $X$ and $Y$ be locally compact topological spaces.
Question 2: Are there non-homeomorphic $X$ and $Y$ such that $M_2\big(C_0(X)\big)$ and $M_2\big(C_0(Y)\big)$ are $\ast$-isomorphic $C^\ast$-algebras?
It is obvious that "2" is not a forced condition.