Let $R$ be a unital ring and let $n\ge 1$ be an integer. Put $A:=M_n(R)$, and for $1\le i,j\le n$ let $e_{ij}\in A$ denote the matrix unit with $1_R$ in the $(i,j)$-entry and $0_R$ elsewhere. Let $e:=e_{11}$. For a unital ring $B$, define $K_0(B)$ as the group completion of the commutative monoid of isomorphism classes of finitely generated projective left $B$-modules under direct sum. Define

\begin{align*} K_1(B):=GL(B)/E(B). \end{align*}

Here $GL(B)=\varinjlim_m GL_m(B)$ under the stabilization maps $C\mapsto\operatorname{diag}(C,1)$, and $E(B)\trianglelefteq GL(B)$ is the stable elementary subgroup generated by all elementary transvections $I_m+bE_{ij}$ with $m\ge 2$, $i\ne j$, and $b\in B$. Then the standard Morita equivalence between $A$ and $R$, given by $M\mapsto eM$ and $N\mapsto Ae\otimes_R N$, induces a natural group isomorphism

\begin{align*} K_0(M_n(R))\cong K_0(R). \end{align*}

The block expansion isomorphisms $M_m(M_n(R))\cong M_{mn}(R)$ identify the corresponding stable general linear groups and stable elementary subgroups, and therefore induce a natural group isomorphism

\begin{align*} K_1(M_n(R))\cong K_1(R). \end{align*}

Equivalently, for $i\in\{0,1\}$ there is a natural isomorphism $K_i(M_n(R))\cong K_i(R)$, natural in the unital ring $R$.