# How are $M_n(A)$ and $M_n(\mathbb{C})\otimes A$ identified as Banach algebras?

When $A$ is a Banach algebra, one can make $M_n(A)$ into a Banach algebra by equipping it with various norms. One can also make the algebraic tensor product $M_n(\mathbb{C})\otimes A$ into a Banach algebra. If we forget about topology, then it seems rather simple to identify $M_n(A)$ and $M_n(\mathbb{C})\otimes A$ algebraically. If we take the norms into account, then usually how are the two identified as Banach algebras? (Included in this question is what norms do one usually consider when making this identification?)

$(a_{ij})→\sigma_i \sigma_j a_{ij} \otimes e_{ij}$ where $e_{ij}$ are the unit matrices