I am trying to understand direct limits in the category of $C^*$ algebras by self reading. My last question was also related to direct limit. Here is my another doubt:

let $(A_n,f_n)$ be a direct sequence of $C^*$ algebras. Does the direct limit behaves well with matrices i.e. $$\lim_{\rightarrow} M_2(A_n)=M_2(\lim_{\rightarrow} A_n)$$ Where for the system $M_2(A_n)$ the connecting maps are natural maps obtained using $f_n$ componentwise.

I do feel like the result should be true but I don’t really have an argument. Any ideas?

I am trying to understand direct limits in the category of $C^*$-algebras by self reading. My last question was also related to direct limits. Here is another of my doubts:

Let $(A_n,f_n)$ be a direct sequence of $C^*$-algebras. Does the direct limit behave well with matrices i.e. $$\lim_{\rightarrow} M_2(A_n)=M_2(\lim_{\rightarrow} A_n)$$ Where for the system $M_2(A_n)$ the connecting maps are the natural maps obtained using $f_n$ componentwise.

I do feel like the result should be true, but I don’t really have an argument. Any ideas?