Proposition 41.3.1.label Let $A$ be a $C^{*}$-algebra, $M_{n}(A) = M_{n}(\complex) \otimes A$ be the space of $n \times n$ matrices over $A$,
\[M_{n}(A) \times M_{n}(A) \to M_{n}(A) \quad (M \otimes a)(N \otimes b) = MN \otimes ab\]
and the involution
\[*: M_{n}(A) \to M_{n}(A) \quad M \otimes a \mapsto M^{*} \otimes a^{*}\]
then there exists a unique norm on $M_{n}(A)$ such that $M_{n}(A)$ is a $C^{*}$-algebra.
Proof. After applying a unitisation, assume without loss of generality that $A$ is unital.
By the Gelfand-Naimark Theorem, there exists a faithful representation $(H, \pi)$ of $A$. Under the identification that $H^{n} = \complex^{n} \otimes H$, $\pi$ induces a faithful unital *-representation
\[\pi_{n}: M_{n}(A) \to B(H^{n}) \quad \pi_{n}(M \otimes a)(\lambda \otimes \xi) = (M\lambda) \otimes (\pi(a)\xi)\]
whose image is a uniformly closed subalgebra of $B(H^{n})$. As such, $B(H^{n})$ induces a norm on $M_{n}(A)$, making it a $C^{*}$-algebra.
By Corollary 38.4.5, this norm is unique.$\square$
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