= Solution
In terms of the <Pauli matrices>, $\gamma^0=\sigma_3$, $\gamma^1=i\sigma_1$, $\gamma^2=i\sigma_2$. The <Pauli matrix multiplication law> implies $\sigma_j^2=I_2$ and $\sigma_j\sigma_k=-\sigma_k\sigma_j$ for $j\ne k$. Consequently
$$
(\gamma^0)^2=I_2,\qquad (\gamma^1)^2=(\gamma^2)^2=-I_2,
\qquad \gamma^\mu\gamma^\nu=-\gamma^\nu\gamma^\mu\quad(\mu\ne\nu).
$$
These diagonal and off-diagonal cases are exactly the <Clifford algebra> relation
$$
\boxed{\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}I_2.}
$$
The identity matrix is implicit on the right-hand side in the source. Useful conventions for the remaining parts are $(\gamma^0)^\dagger=\gamma^0$, $(\gamma^j)^\dagger=-\gamma^j$ for $j=1,2$, and $\gamma^{0*}=\gamma^0$, $\gamma^{1*}=-\gamma^1$, $\gamma^{2*}=\gamma^2$.
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