Let $R$ be a unital, not necessarily commutative, ring, let $n \geq 1$, and put $A:=M_n(R)$. For every unital ring $C$, define $GL(C):=\varinjlim_q GL_q(C)$ under the stabilization maps $X\mapsto \operatorname{diag}(X,1_C)$, let $E(C)\leq GL(C)$ be the stable elementary subgroup, and define $K_1(C):=GL(C)/E(C)$. Let $e=e_{11}(1_R)\in A$, and identify $eAe$ with $R$ by $r\mapsto e_{11}(r)$. Then block expansion $M_q(M_n(R))\cong M_{qn}(R)$ induces an isomorphism of abelian groups $K_1(M_n(R))\cong K_1(R)$. Its inverse is induced by the corner-stabilization map which sends a matrix $X=(x_{ij})\in GL_q(R)$ to the matrix over $A$ acting as $X$ on the $e$-summand and as the identity on the complementary summands. This is the $K_1$ isomorphism associated to the standard Morita idempotent $e$.