Can the existence of antimatter be inferred from Matrix Mechanics?

It is well known that Antimatter was first predicted by interpreting the matrices that show up in the Dirac Equation as indicating its existence.

Dirac factorizes $E^2=p^2+m^2$ ($c=1,\hbar=1$) into $E=\alpha_x \hat p_x+\alpha_y \hat p_y+\alpha_z \hat p_z+\beta m$, such that $$i\partial_t \psi=-i\alpha_x \partial_x\psi-i\alpha_y \partial_y\psi-i\alpha_z \partial_z\psi+\beta m$$ This is only possible if the $\alpha$'s and the $\beta$ are matrices, so the wavefunction is a vector, which implies spin is integrated into the wavefunction, but the matrices have to be $4\times4$ matrices, so the wavefunction is a $4$-component vector, with two solutions of negative energy.

This all happens because we try to find the wavefunction of a relativistic particle. Matrix Mechanics, Heisenberg's formulation, instead of wavefunctions and differential operators has the observables represented by matrices, and the state represented by a state vector. Can spin and antimatter be predicted from this formulation of Quantum mechanics, without starting by accounting for the states of different spin and antimatter in the Hamiltonian a priori?