Help with a step in a derivation in Peskin & Schroeder

In page 53 of P&S, they postulate the commutation relations for the annihilation and creation operators of spin 1/2 particles. From there, they start computing the commutation relations of $psi$ and $psi^{dagger}$. But I don’t get how they arrive at $$ [a_p^r,a_q^{sdagger}]u^r(p)u^{dagger}(q)+[b_{-p}^r,b_{-q}^{sdagger}]v^r(-p)v^{dagger}(-q)$$ (I omitted the rest because I think it’s obvious).
It seems like they are using that $u$ and $u^{dagger}$ commute, but it doesn’t make sense since one product is a scalar ($uu^{dagger}$) and the other a matrix ($u^{dagger}u$). How do they get to the equation I wrote down? I do get the rest of the derivation from there, just not that particular step.

Ps: I know that the operators for spin 1/2 particles anticommute, they just do this as a pedagogical example.