Why in every textbook when deriving Dirac's equation the smallest possible matrices ($2 \times 2$) are used? I wonder why one couldn't use spin 1 matrices ($3 \times 3$) and get relativistic equation for spin 1 particle?
One can write down equations for spin-one particles but they won't be the Dirac equation because the Dirac equation, by definition, describes spin-1/2 particles (i.e. fermions such as leptons and quarks). The equations for free fields associated with spin-one particles are the Maxwell equations (for photons etc.) and/or Proca equations (for W-bosons or Z-bosons).
Particles in the standard model are in principle massless and acquire mass through interactions via the Lagrangian. This poses a problem if you want to use the 3x3 matrix generators of angular momentum because massless particles have different lie group representations as particles with mass.
The 3x3 generators for particles of spin 1 acts on three states, -1, 0 +1. In the massless case only two states remain.
With fermions and the 2x2 SU(2) matrices this issue doesn't exist. The two chiral components $\psi_L$ and $\psi_R$ propagate lightlike and the two states of each chiral component correspond with spin up and down where the spin is parallel or anti parallel to the direction of propagation.