Solution
= Solution
Insert $s_1s_i$ into the spin-basis expansion of the trace. Summing the spins along the first path from site $1$ to site $i$ produces $(M^{i-1})_{st}$, while summing along the complementary path around the periodic chain produces $(M^{N-i+1})_{ts}$. Therefore the <two-point correlation function> is
$$
\boxed{\langle s_1s_i\rangle
=\frac1Z\sum_{s,t=\pm1}st
(M^{i-1})_{st}(M^{N-i+1})_{ts}}.
$$