I would like to show the following isomorphy but not sure how to go about this: $\mathbb{K}\cong M_{n}(\mathbb{K})$ Also in Blackadar (Operator Algebras, page 171) he states that this isomorphism induces: $B\cong M_{2}(B)$, $M(B)\cong M_{2}(M(B))$ and $Q(B)\cong M_{2}(Q(B))$, which he calls standard isomorphisms.<br> In the context or the present question, the notation has the following meaning: * $B$ is a $C^\ast$-algebra, * $Q$ is the outer multiplier algebra of $B$ (hence $Q = M(B)/B$), * $M$ is the multiplier algebra, * $M_{n}(\mathbb{K})$ are $n\times n$ matrices with entries from $\mathbb{K}$, and finally * $\mathbb{K}$ is the $C^\star$-algebra of compact operators on a separable, infinite-dimensional Hilbert space. Anyway, I don't know how to show this and cannot find a reference on how to do this. Would be very thankful for a reference, how to do it or even some hint.