When trying to derive the probability density from the Pauli equation, I face a problem. Starting from the Pauli equation $$ i\hbar \frac{\partial \Psi}{\partial t}=\hat H_0 \Psi +\mu_B \ \hat \sigma \cdot \mathbf{B} \Psi, $$ I need to adjoint it: $$ -i\hbar \frac{\partial \Psi^+ }{\partial t}=\hat H_0^* \Psi^+ +\mu_B \ \left( \hat \sigma \cdot \mathbf{B} \Psi \right)^+. $$ So, I'm trying to calculate the multiplication in brackets, using the properties of conjugation $(AB)^+=B^+A^+$: $$ \left( \hat \sigma \cdot \mathbf{B} \Psi \right)^+ \equiv \bigg( \left( \hat \sigma \cdot \mathbf{B} \right) \Psi \bigg)^+= \Psi^+ \left( \hat \sigma \cdot \mathbf{B} \right)^+ = \Psi^+ \mathbf{B}^+ \hat \sigma^+= \Psi^+ \mathbf{B}^T \hat \sigma. $$ Here I've used the facts that the magnet field is real $(\mathbf{B}^+=\mathbf{B}^T)$ and pauli matrices are hermitian $(\hat \sigma^+ =\hat \sigma)$.
However, in the book (Greiner, Quantum Mechanics: an introduction) there is another answer: $$ \left( \hat \sigma \cdot \mathbf{B} \Psi \right)^+ = \Psi^+ \hat \sigma \cdot \mathbf{B}. $$ Where is the mistake? Thanks in advance.
P.S. I understand that $\hat \sigma$ is an operator, and so it must act some function, but... I still don't see my mistake.