= Solution
The cyclic products obtained by direct <matrix multiplication> are $\sigma_x\sigma_y=i\sigma_z$, $\sigma_y\sigma_z=i\sigma_x$, and $\sigma_z\sigma_x=i\sigma_y$. Reversing any two distinct indices changes the sign; equal-index products are $I$. Therefore every pair is covered by
$$
\boxed{\{\sigma_i,\sigma_j\}=2\delta_{ij}I,\qquad
[\sigma_i,\sigma_j]=2i\sum_k\epsilon_{ijk}\sigma_k.}
$$
Here $\{A,B\}$ is the <anticommutator> and $[A,B]$ the <commutator>. Explicitly the three diagonal <anticommutators> are $2I$, all six off-diagonal ones are zero, and all three diagonal <commutators> vanish. The cyclic <commutators> are $2i\sigma_z,2i\sigma_x,2i\sigma_y$, respectively, with their reversed partners negative. The convention is $\epsilon_{xyz}=1$.
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