 Solution

The Clifford algebra condition for signature $(-,+,+)$ isEach Pauli matrix squares to $I_2$, and distinct Pauli matrices anticommute. Therefore the given choice obeysThese check all diagonal and off-diagonal anticommutators, provingThus they are viable three-dimensional gamma matrices. Their Hermitian adjoints satisfy ${\gamma }_0^{†}=-{\gamma }_0$ and ${\gamma }_a^{†}={\gamma }_a$ for $a=1,2$. Two components are minimal: one-dimensional nonzero scalar matrices cannot anticommute while all having nonzero squares.