Complex coefficents in density operator expansion?

[center o bass]

Hey, I recently had an exam where the quantum state were on the form

$$|\psi\rangle = \frac{1}{\sqrt{2}} ( |+\rangle + i |-\rangle )$$

Here I formed the density operator for the pure state

$$\rho(t) = |\psi\rangle \langle \psi| = \frac{1}{2} ( |+\rangle + i |-\rangle )( \langle +| - i \langle -| ) = \frac{1}{2} ( |+\rangle \langle + | + |- \rangle \langle - | + i(|-\rangle \langle + | - |+\rangle \langle -|)).$$

However in the solutions for the exam the complex i's were not there, i.e the solutions states that

$$\rho = \frac{1}{2} ( |+\rangle \langle + | + |- \rangle \langle - | + |-\rangle \langle +| - |+\rangle \langle - |).$$

Have I missed something here or is the suggested solution erroneous? Is there a reason why a density operator expansion should not have complex coefficients?

 

[Jano L.]

Your solution seems correct - I've got i's on anti-diagonal as well. The second expression cannot be right, because the density matrix has to be hermitian.