= Solution
The <Clifford algebra> condition for signature $(-,+,+)$ is
$$
\{\gamma_\mu,\gamma_\nu\}=2\eta_{\mu\nu}I_2.
$$
Each <Pauli matrix> squares to $I_2$, and distinct <Pauli matrices> anticommute. Therefore the given choice obeys
$$
\gamma_0^2=(-i\sigma_1)^2=-I_2,\qquad
\gamma_1^2=\gamma_2^2=I_2,\qquad
\gamma_\mu\gamma_\nu+\gamma_\nu\gamma_\mu=0\quad(\mu\ne\nu).
$$
These check all diagonal and off-diagonal <anticommutators>, proving
$$
\boxed{\{\gamma_\mu,\gamma_\nu\}=2\operatorname{diag}(-1,1,1)_{\mu\nu}I_2.}
$$
Thus they are viable three-dimensional <gamma matrices>. Their <Hermitian adjoints> satisfy $\gamma_0^\dagger=-\gamma_0$ and $\gamma_a^\dagger=\gamma_a$ for $a=1,2$. Two components are minimal: one-dimensional nonzero scalar <matrices> cannot anticommute while all having nonzero squares.
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