Questions about multiplier algebra and corona algebra When I read N.E. Wegge-Olsen's book K-theory and C-star-algebras_ A friendly approach I meet the following two problems about standard isomophisms: 1. For any $C^\ast$-algebra $\mathcal{A}$, is $\mathbb{M}_n(\cal C({A}))\cong C(\mathbb{M}_n(\cal{A}))$? where $\cal{C(A)}$ means the corona algebra $M(\cal{A})/\cal{A}$ and $M(\cal{A})$ means the multiplier algebra of $A$. $\mathbb{M}_n(A)$ means the algebra of $n\times n$ matrices with coefficients in $\cal{A}$. 2. How dose an isomorphism from $\mathbb{M}_2(\cal{A}\otimes\cal{K(H)})$ to $\cal{A}\otimes\cal{K(H)}$ induce an isomorphism from $\mathbb{M}_2(\cal{C}(\cal{A}\otimes\cal{K(H)}))$ to $\cal{C}(\cal{A}\otimes\cal{K(H)})$? where $\cal{K(H)}$ means the algebra of all compact operators on separable Hilbert space $\cal{H}$. Thanks a lot!

It is not hard to check that $M(\mathbb M_n(\mathcal A))=\mathbb M_n(M(\mathcal A))$, and that $\mathbb M_n(\mathcal A)/\mathbb M_n(\mathcal J)=M_n(\mathcal A/\mathcal J)$ for any C$^*$-algebra $\mathcal A$ with ideal $\mathcal J$. Then $$ \mathbb M_n(M(\mathcal A)/\mathcal A)=\mathbb M_n(M(\mathcal A))/\mathbb M_n(\mathcal A)=M(\mathbb M_n(\mathcal A))/\mathbb M_n(\mathcal A). $$ That shows part 1. For part 2, we just apply part 1, and the isomorphism given: $$ \mathbb{M}_2(\cal{C}(\cal{A}\otimes\cal{K(H)}))=\mathcal C(\mathbb{M}_2(\cal{A}\otimes\cal{K(H)})\simeq \mathcal C(\cal{A}\otimes\cal{K(H)}) $$