I want to know if gamma matrices commute with the Dirac spinor field, i.e., are the following equalities correct?
$$ \psi\gamma^{\mu}\overset{?}{=}\gamma^{\mu}\psi $$
$$ \psi^{\dagger}\gamma^{\mu}\overset{?}{=}\gamma^{\mu}\psi^{\dagger} $$
$$ \overline{\psi}\gamma^{\mu}=\psi^{\dagger}\gamma^{0}\gamma^{\mu}=\psi^{\dagger}\left(2\eta^{0\mu}-\gamma^{\mu}\gamma^{0}\right)\overset{?}{=}2\eta^{0\mu}\psi^{\dagger}-\gamma^{\mu}\overline{\psi} $$