 Solution

Besides their squares, direct multiplication yields ${\sigma }_x{\sigma }_y=i{\sigma }_z$, ${\sigma }_y{\sigma }_z=i{\sigma }_x$ and ${\sigma }_z{\sigma }_x=i{\sigma }_y$. Reversing each pair changes the sign. These identities combine into the Pauli matrix multiplication identityAdding the expression with $i,j$ exchanged cancels the antisymmetric term, proving $\{{\sigma }_i,{\sigma }_j\}=2{\delta }_{ij}I$. Taking the trace instead uses $\mathrm{tr}I=2$ and $\mathrm{tr}{\sigma }_k=0$, proving $\mathrm{tr}({\sigma }_i{\sigma }_j)=2{\delta }_{ij}$.