Prove this is an equivalence relation

Question

$A$ is related to $B$ if $M_n(A)\simeq M_m(B)$ for some integers $m$ and $n$.

Clearly reflexivity and symmetry are trivial. It's transitivity that I am struggling with. Is it the case that if $M_n(A)\simeq M_m(B)$ then $M_{nk}(A) \simeq M_{mk}(B)$? If not then what can I do?

Thanks.

Answer 1

It is the case.

Note that $M_{nk}(A) \cong M_{kn}(A) \cong M_{k}(M_n(A)) \cong M_k(M_m(B)) \cong M_{mk}(B)$, the part $M_{nk}(A) = M_n(M_k(A))$ is why block notation works. i.e writing a matrix in $M_{2n}(R)$ as $\begin{pmatrix} A & B \\ C & D \end{pmatrix}$ where $A,B,C,D \in M_n(R)$ for some ring $R$.