  Let $R$ denote the purely infinite simple unital Leavitt path algebra $L(E)$.
We completely determine the pairs of positive integers $(c,d)$ for which there
is an isomorphism of matrix rings $M_c(R)\cong M_d(R)$, in terms of the order
of $[1_R]$ in the Grothendieck group $K_0(R)$.
